%I A000041 M0663 N0244
%S A000041 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,
%T A000041 1002,1255,1575,1958,2436,3010,3718,4565,5604,6842,8349,10143,12310,
%U A000041 14883,17977,21637,26015,31185,37338,44583,53174,63261,75175,89134
%N A000041 a(n) = number of partitions of n (the partition numbers).
%C A000041 Also number of nonnegative solutions to b+2c+3d+4e+...=n and the number
of nonnegative solutions to 2c+3d+4e+...<=n. - Henry Bottomley (se16(AT)btinternet.com),
Apr 17 2001
%C A000041 a(n) is also the number of conjugacy classes in the symmetric group S_n
(and the number of irreducible representations of S_n).
%C A000041 Also the number of rooted trees with n+1 nodes and height at most 2.
%C A000041 Coincides with the sequence of numbers of nilpotent conjugacy classes
in the Lie algebras gl(n). A006950, A015128 and this sequence together
cover the nilpotent conjugacy classes in the classical A,B,C,D series
of Lie algebras. - Alexander Elashvili, Sep 08 2003
%C A000041 a(n)=a(0)b(n)+a(1)b(n-2)+a(2)b(n-4)+... where b=A000009.
%C A000041 Number of distinct Abelian groups of order p^n, where p is prime (the
number is independent of p). - Lekraj Beedassy (blekraj(AT)yahoo.com),
Oct 16 2004
%C A000041 Number of graphs on n vertices that do not contain P3 as an induced subgraph.
- Washington Bomfim (webonfim(AT)bol.com.br), May 10 2005
%C A000041 It is unknown if there are infinitely many partition numbers divisible
by 3, although it is known that there are infinitely many divisible
by 2. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 21 2005
%C A000041 Numbers of terms to be added when expanding the n-th derivative of 1/
f(x). - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov
07 2005
%C A000041 a(n) = A114099(9*n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Feb 15 2006
%C A000041 Comment from Maurice D. Craig (towenaar(AT)optusnet.com.au), Oct 30 2006:
sequence agrees with expansion of Molien series for symmetric group
S_n up to the term in x^n.
%C A000041 Also the number of nonnegative integer solutions to x_1+x_2+x_3+...+x_n=n
such that n>=x_1>=x_2>=x_3>=...>=x_n>=0, because by letting y_k=x_k-x_(k+1)>
=0 (where 0<k<n) we get y_1+2y_2+3y_3+...+(n-1)y_(n-1)+nx_n=n. -
Werner Grundlingh (wgrundlingh(AT)gmail.com), Mar 14 2007
%C A000041 Let P(z):= Sum{j=0..inf} b_j z^j, b_0 != 0. Then 1/P(z) = Sum{j=0..inf}
c_j z^j, where the c_j must be computed from the infinite triangular
system b_0 c_0 = 1, b_0 c_1 + b_1 c_0 = 0 and so on (Cauchy products
of the coefficients set to zero). The n-th partition number arises
as the number of terms in the numerator of the expression for c_n:
The coefficient c_n of the inverted power series is a fraction with
b_0^(n+1) in the denominator and in its numerator having a(n) products
of n coefficients b_i each. The partitions may be read off from the
indices of the b_i. - Peter C. Heinig (algorithms(AT)gmx.de), Apr
09 2007
%C A000041 A026820(a(n),n) = A134737(n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Nov 07 2007
%C A000041 Equals row sums of triangle A137683 - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Feb 05 2008
%C A000041 This is also the number of parts equal to 1 in the outer shell of the
partitions of n+1 (see A138151). - Omar E. Pol (info(AT)polprimos.com),
Apr 17 2008
%C A000041 a(n)= the number of different ways to run up a staircase with n steps,
taking steps of sizes 1,2,3,... and r (r<=n), where the order is
not important and there is no restriction on the number or the size
of each step taken. - Mohammad K. Azarian (azarian(AT)evansville.edu),
May 21 2008
%C A000041 Equals the eigenvector of triangle A145006 and row sums of the eigentriangle
of the partition numbers, A145007. [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Sep 28 2008]
%C A000041 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 05 2008:
(Start)
%C A000041 Starting with offset 1 = INVERT transform of (1, 1, 0, 0, -1, 0, -1,...),
%C A000041 where A080995, the characteristic function of A001318 (1, 2, 5, 7, 12,
...) is
%C A000041 signed (++ -- ++,...) as to 1's. This is equivalent to Lim__{n=1..inf}
%C A000041 A145006^n as a vector. The INVERT transform of (1, 1, 0, 0, -1,...) begins
(1, 2,..)
%C A000041 then for each successive operation we take a dot product of (1, 1, 0,
0, -1,...) in reverse and the ongoing results of our series (1, 2,
3, 5, 7,...)
%C A000041 then add the result to the next term in (1, 1, 0, 0, -1,...). For example,
a
%C A000041 (7) = 15 = (0, -1, 0, 0, 1, 1) dot (1, 2, 3, 5, 7, 11) = (0*1, (-1)*2,
0*3, 0*5, 1*7, 1*11)
%C A000041 = (-2 + 7 + 11) = 16, then add to (-1) = 15. (End)
%C A000041 Convolved with A147843 = A000203 prefaced with a zero: (0, 1, 3, 4, 7,
...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2008]
%C A000041 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 12 2009:
(Start)
%C A000041 Equals an infinite convolution product_(1,1,1,...)*(1,0,1,0,1,...)*
%C A000041 (1,0,0,1,0,0,1,...)*(1,0,0,0,1,0,0,0,1,...)* ...; = a*b*c*...; where
a =
%C A000041 (1/(1-x), b = (1/(1-x^2), c = (1/(1-x^3), ...etc. An array by rows: row
1 =
%C A000041 a, row 2 = a*b, row 3 = a*b*c,...; gives:
%C A000041 1, 1, 1, 1, 1, 1,. 1,. 1,. 1,..1,... = (a).................................
%C A000041 1, 1, 2, 2, 3, 3,. 4,..4,. 5,..5,... = (a*b)...............................
%C A000041 1, 1, 2, 3, 4, 5,. 7,..8,.10,.11,... = (a*b*c).............................
%C A000041 1, 1, 2, 3, 4, 5,. 6,..9,.11,.17,... = (a*b*c*d)...........................
%C A000041 1, 1, 2, 3, 5, 5,. 7,.10,.13,.18,... = (a*b*c*d*e).........................
%C A000041 1, 1, 2, 3, 5, 7,.11,.14,.20,.25,... = (a*b*c*d*e*f).......................
%C A000041 1, 1, 2, 3, 5, 7,.11,.15,.21,.27,... = (a*b*c*d*e*f*g).....................
%C A000041 1, 1, 2, 3, 5, 7,.11,.15,.22,.28,... = (a*b*c*d*e*f*g*h)...................
%C A000041 1, 1, 2, 3, 5, 7,.11,.15,.22,.29,... = (a*b*c*d*e*f*g*h*i).................
%C A000041 ...with rows tending to A000041. Partition triangles A058398 = ascending
%C A000041 antidiagonals. Partition triangle A008284 reversal of A058398. (End)
%C A000041 a(n) is also the number of partitions of 2n into even parts. More generally,
it appears that a(n) is also the number of partitions of k*n into
parts divisible by k, for k>0 and n>0. [From Omar E. Pol (info(AT)polprimos.com),
Nov 20 2009]
%D A000041 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, 1964 (and various
reprintings), p. 836.
%D A000041 George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading,
Mass., 1976
%D A000041 G. E. Andrews & K. Ericksson, Integer Partitions, Cambridge University
Press 2004.
%D A000041 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 307.
%D A000041 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math.
Soc., 1963; Chapter III.
%D A000041 Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem,
Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28,
Winter 1997
%D A000041 B. C. Berndt, Number Theory in the Spirit of Ramanujan, Chap. I Amer.
Math. Soc. Providence RI 2006.
%D A000041 L. E. Dickson, History of the Theory of Numbers, Vol.II Chapter III pp.
101-164,Chelsea NY 1992.
%D A000041 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math.
Soc., 1988; p. 37, Eq. (22.13).
%D A000041 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables,
Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D A000041 G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatorial analysis,
Proc. London Math. Soc., 17 (1918), 75-.
%D A000041 J. M. Kane, Distribution of orders of Abelian groups, Math. Mag., 49
(1976), 132-135.
%D A000041 D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial
Algorithms, (to appear), section 7.2.1.4.
%D A000041 S. Markovski and M. Mihova, An explicit formula for computing the partition
numbers p(n), preprint, 2005
%D A000041 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.1,
p. 491.
%D A000041 S. Ramanujan, Collected Papers, Chap. 25, Cambridge Univ. Press 1927
(Proceedings of the Camb.Phil.Soc., 19(1919)207-213).
%D A000041 S. Ramanujan, Collected Papers, Chap. 28, Cambridge Univ. Press 1927
(Proceedings of the London Math.Soc., 2, 18(1920)).
%D A000041 S. Ramanujan, Collected Papers, Chap. 30, Cambridge Univ. Press 1927
(Mathematische Zeitschrift, 9(1921)147-163).
%D A000041 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p.
122.
%D A000041 J. Riordan, Enumeration of trees by height and diameter, IBM J. Res.
Dev. 4 (1960), 473-478.
%D A000041 J. E. Roberts, Lure of the Integers, pp. 168-9 MAA 1992.
%D A000041 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000041 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000041 R. E. Tapscott and D. Marcovich, "Enumeration of Permutational Isomers:
The Porphyrins", Journal of Chemical Education, 55 (1978), 446-447.
[From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Sep 21 2008]
%D A000041 Robert M. Young, "Excursions in Calculus", Mathematical Association of
America, p. 367. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct
05 2008]
%D A000041 Robert M. Ziff, "On Cardy's formula for the critical crossing probability
in 2d percolation," J. Phys. A. 28, 1249-1255 (1995).
%H A000041 David W. Wilson, <a href="b000041.txt">Table of n, a(n) for n = 0..10000</
a>
%H A000041 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A000041 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%H A000041 S. Ahlgren and K. Ono, <a href="http://www.ams.org/notices/200109/fea-ahlgren.pdf">
Addition and Counting: The Arithmetic of Partitions</a>
%H A000041 S. Ahlgren & K. Ono, <a href="http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=60793">
Congruence properties for the partition function</a>
%H A000041 S. Ahlgren & K. Ono, <a href="http://www.pnas.org/cgi/content/full/98/
23/12882">Congruence properties for the partition function</a>
%H A000041 G. Almkvist, <a href="http://www.expmath.org/restricted/7/7.4/almkvist.ps">
Asymptotic Formulas and Generalized Dedekind Sums</a>
%H A000041 G. Almkvist and H. S. Wilf, <a href="http://citeseer.nj.nec.com/correct/
513487">On the coefficients in the Hardy-Ramanujan-Rademacher formula
for p(n)</a>
%H A000041 Amazing Mathematical Object Factory, <a href="http://www.aarms.math.ca/
ACMN/amof/e_partI.htm">Information on Partitions</a> [Broken link
corrected by Steve Vonn (5463math(AT)gmail.com), Jan 03 2009]
%H A000041 G. E. Andrews, <a href="http://www.emis.de/journals/SLC/opapers/s25andrews.pdf">
Three Aspects of Partitions</a>
%H A000041 G. E. Andrews, <a href="http://www.combinatorics.org/Volume_11/PDF/v11i2r1.pdf">
On a Partition Function of Richard Stanley</a>.
%H A000041 G. E. Andrews & K. Ono, <a href="http://pubmedcentral.com/articlerender.fcgi?artid=1266147">
Ramanujan's congruences and Dyson's crank</a>
%H A000041 G. E. Andrews & R. Roy, <a href="http://www3.combinatorics.org/Volume_4/
PDF/v4i2r02.pdf">Ramanujan's Method in q-series Congruences</a>
%H A000041 Anonymous, <a href="http://felix.unife.it/Root/d-Mathematics/d-Number-theory/
b-Partitions">Bibliography on Partitions</a>
%H A000041 A. O. L. Atkins & F. G. Garvan, <a href="http://arXiv.org/abs/math.NT/
0208050">Relations between the ranks and cranks of partitions</a>
%H A000041 A. Berkovich & F. G. Garvan, <a href="http://arXiv.org/abs/math.CO/0401012">
On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence
Modulo 5</a>
%H A000041 A. Berkovich & F. G. Garvan, <a href="http://arXiv.org/abs/math.CO/0402439">
On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence
Modulo 5 and Generalizations</a>
%H A000041 B. C. Berndt, <a href="http://www.math.uiuc.edu/~berndt/articles/partitions2.pdf">
Ramanujan's congruences for the partition function modulo 5,7 and
11</a>
%H A000041 B. C. Berndt and K. Ono, <a href="http://www.math.wisc.edu/~ono/reprints/
044.pdf">Ramanujan's Unpublished Manuscript On The Partition And
Tau Functions With Proofs And Commentary</a>
%H A000041 B. C. Berndt and K. Ono, <a href="http://emis.dsd.sztaki.hu/journals/
SLC/wpapers/s42berndt.html">Ramanujan's Unpublished Manuscript on
the Partition and Tau Functions with Proofs and Commentary</a>
%H A000041 H. Bottomley, <a href="a008284.gif">Illustration of initial terms</a>
%H A000041 H. Bottomley, <a href="a9.gif">Illustration of initial terms of A000009,
A000041 and A047967</a>
%H A000041 H. Bottomley, <a href="http://www.btinternet.com/~se16/js/partitions.htm">
Partition and composition calculator</a>
%H A000041 K. S. Brown, <a href="http://www.math.niu.edu/~rusin/known-math/95/partitions">
Additive Partitions of Numbers</a>
%H A000041 K. S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath383.htm">
Computing the Partitions of n</a>
%H A000041 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000041 Huantian Cao, <a href="http://www.cs.uga.edu/~rwr/STUDENTS/hcao.html">
AutoGF: An Automated System to Calculate Coefficients of Generating
Functions</a>.
%H A000041 J. Davis & E. Perez, <a href="http://www.ces.clemson.edu/~kevja/REU/2002/
JDavisAndEPerez.pdf">Computations Of The Partition Function, p(n)</
a>
%H A000041 N. J. Fine, <a href="http://www.pnas.org/cgi/reprint/34/12/616.pdf">Some
New Results On Partitions</a>
%H A000041 B. Forslund, <a href="http://my.tbaytel.net/~forslund/partitio.html">
Partitioning Integers</a>
%H A000041 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/
k1partn.html">Partitions of an Integer</a>
%H A000041 GEO magazine, <a href="http://www.geo.de/GEO/wissenschaft_natur/technik/
2000_11_GEO_11_zahlenspalterei/">Zahlenspalterei</a>
%H A000041 A. Hassen and T. J. Olsen, <a href="http://www.math.temple.edu/~melkamu/
html/partition.pdf">Playing With Partitions On The Computer</a>
%H A000041 A. D. Healy, <a href="http://www.alexhealy.net/papers/math192.pdf">Partition
Identities</a>
%H A000041 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=61">
Encyclopedia of Combinatorial Structures 61</a>
%H A000041 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=74">
Encyclopedia of Combinatorial Structures 74</a>
%H A000041 E. Klarreich, <a href="http://www.sciencenews.org/articles/20050618/bob9.asp">
Pieces of Numbers: A proof brings closure to a dramatic tale of partitions
and primes</a>, Science News, Week of Jun 18, 2005; Vol. 167, No.
25, p. 392.
%H A000041 J. Laurendi, <a href="http://www.artofproblemsolving.com/Resources/Papers/
LaurendiPartitions.pdf">Partitions of Integers</a>
%H A000041 T. Lockette, Explore Magazine, <a href="http://rgp.ufl.edu/explore/v05n2/
math.html">"Path To Partitions"</a>
%H A000041 Dr. Math, <a href="http://mathforum.org/dr.math/problems/partitions.html">
Partitioning the Integers</a>
%H A000041 Dr. Math, <a href="http://mathforum.org/dr.math/problems/huckin11.14.98.html">
Partitioning an Integer</a>
%H A000041 M. MacMahon, Collected Papers of Ramanujan, <a href="http://www.imsc.res.in/
~rao/ramanujan/CamUnivCpapers/Cpaper36/page33.htm">Table for p(n);
n=1 through 200</a>
%H A000041 G. P. Michon, <a href="http://home.att.net/~numericana/data/partition.htm">
Table of partition function p(n) (n=0 through 4096)</a>
%H A000041 G. P. Michon, <a href="http://home.att.net/~numericana/answer/numbers.htm#partitions">
Partition function</a>
%H A000041 G. A. Miller, <a href="http://www.pnas.org/cgi/reprint/22/11/654.pdf">
Number Of The Abelian Groups Of A Given Order</a>
%H A000041 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
part">Factorization of Partition Numbers</a>
%H A000041 D. J. Newman, <a href="http://projecteuclid.org/Dienst/UI/1.0/Summarize/
euclid.mmj/1028998729">A Simplified Proof Of The Partition Formula</
a>
%H A000041 K. Ono, <a href="http://math.la.asu.edu/~sf2000/kono.pdf">Arithmetic
of The Partition Function</a>
%H A000041 K. Ono, <a href="http://www.ams.org/era/1995-01-01/S1079-6762-95-01005-5/
S1079-6762-95-01005-5.pdf">Parity Of The Partition Function</a>
%H A000041 K. Ono, <a href="http://www.emis.de/journals/Annals/151_1/ono.pdf">Distribution
of the partition function modulo m</a>
%H A000041 T. J. Osler, <a href="http://www2.rowan.edu/mars/depts/math/HASSEN/NT/
Playpart.html">Playing with Partitions on the Computer</a>
%H A000041 I. Peterson, <a href="http://www.sciencenews.org/20000617/bob10.asp">
The Power Of Partitions</a>
%H A000041 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting
Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004),
Article 04.3.2.
%H A000041 M. Planat, <a href="http://arXiv.org/abs/math-ph/0307033">Quantum 1/f
Noise in Equilibrium: from Planck to Ramanujan</a>
%H A000041 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/OEIS/A000041">
Partitions</a> [Contains first 10000000 terms]
%H A000041 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/OEIS/A000041_to_300000.txt">
Partition numbers through n = 300000</a>
%H A000041 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/OEIS/A000041_300k_to_450k.txt">
Partitions numbers from 300000 to 450000</a>
%H A000041 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/OEIS/A000041_450k_to_500k.txt">
Partitions numbers from 450000 to 500000</a>
%H A000041 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polparti.jpg">
How to build a shell model of partitions</a> [From Omar E. Pol (info(AT)polprimos.com),
Sep 06 2008]
%H A000041 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">
A shell model of partitions (2D and 3D)</a> [From Omar E. Pol (info(AT)polprimos.com),
Sep 06 2008]
%H A000041 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">
Illustration of initial terms (2D view)</a> [From Omar E. Pol (info(AT)polprimos.com),
Sep 06 2008]
%H A000041 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">
Illustration of initial terms (3D view)</a> [From Omar E. Pol (info(AT)polprimos.com),
Sep 06 2008]
%H A000041 M. Presern, <a href="http://www2.arnes.si/massvega/documents/ke-2003/
Some-Results-on-Partitions.pdf">Some Results On Partitions</a>
%H A000041 W. A. Pribitkin, The Ramanujan Journal 4(4) 2000, <a href="http://www.wkap.nl/
oasis.htm/310442">Revisiting Rademacher's Formula for the Partition
Function p(n)</a>
%H A000041 PYTHAGORAS, <a href="http://www.science.uva.nl/misc/pythagoras/jaargang/
9899/aug99/partities.php3">Ramanujan and The Partition Function(Text
in Dutch)</a>
%H A000041 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
Cpaper25/page1.htm">Some Properties Of p(n), The Number Of Partitions
Of n</a>
%H A000041 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
Cpaper28/page1.htm">Congruence Properties Of Partitions</a>
%H A000041 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
Cpaper30/page1.htm">Congruence Properties Of Partitions</a>
%H A000041 S. Ramanujan & G. H. Hardy, <a href="http://www.imsc.res.in/~rao/ramanujan/
CamUnivCpapers/Cpaper31/page1.htm">Une formule asymptotique pour
le nombre de partitions de n</a>
%H A000041 J. D. Rosenhouse, <a href="http://www.math.ksu.edu/~jasonr/book4.pdf">
Partitions of Integers</a>
%H A000041 J. D. Rosenhouse, <a href="http://www.math.ksu.edu/~jasonr/Solutions4.pdf">
Solutions to Problems</a>
%H A000041 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/gen/nump.html">
Generate Numerical Partitions</a>
%H A000041 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/tables/partitions.txt.gz">
The first 284547 partition numbers</a> (52MB compressed file)
%H A000041 M. Savic, <a href="http://www.cs.bsu.edu/homepages/fischer/Journal/01-01/
savic.pdf">The Partition Function and Ramanujan's 5k+4 Congruence</
a>
%H A000041 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/
series011">Number of integer partitions</a>
%H A000041 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.ps">
A combinatorial miscellany</a>
%H A000041 R. L. Weaver, The Ramanujan Journal 5(1) 2001, <a href="http://www.wkap.nl/
oasis.htm/323807">New Congruences for the Partition Function</a>
%H A000041 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Partition.html">Link to a section of The World of Mathematics (1).</
a>
%H A000041 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PartitionFunctionP.html">Link to a section of The World of Mathematics
(2).</a>
%H A000041 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
RamanujansIdentity.html">Link to a section of The World of Mathematics(3)</
a>
%H A000041 West Sussex Grid for Learning, Multicultural Mathematics, <a href="http:/
/wsgfl.westsussex.gov.uk/maths/Ramanujan.htm">Ramanujan's Partition
of Numbers</a>
%H A000041 Thomas Wieder, <a href="a000041.txt">Comment on A000041</a>
%H A000041 Wikipedia, <a href="http://www.wikipedia.org/wiki/integer_partition">
Integer Partition</a>
%H A000041 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/PIMS/PIMSLectures.pdf">
Lectures on Integer Partitions</a>
%H A000041 Wolfram Research, <a href="http://functions.wolfram.com/IntegerFunctions/
PartitionsP/11">Generating functions of p(n)</a>
%H A000041 D. J. Wright, <a href="http://www.math.okstate.edu/~wrightd/4713/nt_essay/
node14.html">Partitions</a>
%H A000041 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000041 <a href="Sindx_Par.html#partN">Index entries for related partition-counting
sequences</a>
%H A000041 <a href="Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k
>= 1} (1-x^k)^m</a>
%H A000041 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to
rooted trees</a>
%H A000041 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
Publications/books.html">Analytic Combinatorics</a>, 2009; see page
41
%F A000041 G.f.: Product_{k>0} 1/(1-x^k) = Sum_{k>= 0} x^k Product_{i = 1..k} 1/
(1-x^i) = 1+Sum_{k>0} x^(k^2)/(Product_{i = 1..k} (1-x^i))^2.
%F A000041 a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + ... =
0, where the sum is over n-k and k is a generalized pentagonal number
(A001318) <= n and the sign of the k-th term is (-1)^([(k+1)/2]).
See A001318 for a good way to remember this!
%F A000041 a(n) = (1/n) * Sum_{k=0, 1, ..., n-1} sigma(n-k)*a(k), where sigma(k)
is the sum of divisors of k (A000203).
%F A000041 a(n) ~ 1/(4*n*sqrt(3)) * e^(Pi * sqrt(2n/3)) as n -> infinity (Hardy
and Ramanujan).
%F A000041 a(n) < exp( (2/3)^(1/2) pi sqrt(n) ) (Ayoub, p. 197).
%F A000041 G.f.: Product (1+x^m)^A001511(m); m=1..inf. - Vladeta Jovovic (vladeta(AT)eunet.rs),
Mar 26 2004
%F A000041 a(n)=sum(i=0, n-1, P(i, n-i)), where P(x, y) is the number of partitions
of x into at most y parts and P(0, y)=1. - Jon Perry (perry(AT)globalnet.co.uk),
Jun 16 2003
%F A000041 G.f. : product(i=1, oo, product(j=0, oo, (1+x^((2i-1)*2^j))^(j+1))) -
Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004
%F A000041 G.f. e^{Sum_{k>0} (x^k/(1-x^k)/k)}. - Frank Adams-Watters (FrankTAW(AT)Netscape.net),
Feb 08 2006
%F A000041 Euler transform of all 1's sequence (A000012). Weighout transform of
A001511. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 15
2006
%F A000041 a(n) = A027187(n)+A027193(n) = A000701(n)+A046682(n). - Reinhard Zumkeller
(reinhard.zumkeller(AT)gmail.com), Apr 22 2006
%F A000041 Row sums of triangles A133734 and A133736. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Sep 22 2007
%F A000041 1/(1-x) + sum{ k=1,oo, x^k(k+1)/ prod(i=1,k, (1-x^i)^2)*(1-x^k+1) } (the
pronic equivalent of the Durfee Square GF) [From Jon Perry (johnandruth(AT)jrperry.orangehome.co.uk),
Aug 02 2008]
%F A000041 Convolved with A152537 gives A000079, powers of 2. [From Gary W. Adamson
(qntmpkt(AT)haoo.com), Dec 06 2008]
%p A000041 with(combinat); A000041 := numbpart; [ seq(numbpart(i),i=0..50) ]; [Warning:
Maple 10 and 11 give incorrect answers in some cases, for example
combinat[numbpart](11269); is wrong.]
%p A000041 spec := [ B, {B=Set(Set(Z,card>=1))}, unlabeled ]; [seq(combstruct[count](spec,
size=n), n=0..50)];
%p A000041 with(combstruct):ZL0:=[S,{S=Set(Cycle(Z,card>0))}, unlabeled]:seq(count(ZL0,
size=n),n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Sep 24 2007
%p A000041 G:={P=Set(Set(Atom,card>0))}:combstruct[gfsolve](G,labeled,x);seq(combstruct[count]([P,
G,unlabeled],size=i),i=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Dec 16 2007
%t A000041 Table[ PartitionsP[n], {n, 0, 45}]
%o A000041 (MAGMA) a:= func< n | NumberOfPartitions(n) >; [ a(n) : n in [0..10]];
%o A000041 (PARI) a(n)=if(n<0,0,polcoeff(1/eta(x+x*O(x^n)),n))
%o A000041 (PARI) The Hardy-Ramanujan-Rademacher exact formula in PARI is as follows
(this is no longer necessary since it is now built in to the numbpart
command): - Ralf Stephan (ralf(AT)ark.in-berlin.de), Nov 30 2002
%o A000041 Psi(n, q) = local(a, b, c); a=sqrt(2/3)*Pi/q; b=n-1/24; c=sqrt(b); (sqrt(q)/
(2*sqrt(2)*b*Pi))*(a*cosh(a*c)-(sinh(a*c)/c))
%o A000041 L(n, q) = if(q==1,1,sum(h=1,q-1,if(gcd(h,q)>1,0,cos((g(h,q)-2*h*n)*Pi/
q))))
%o A000041 g(h, q) = if(q<3,0,sum(k=1,q-1,k*(frac(h*k/q)-1/2)))
%o A000041 part(n) = round(sum(q=1,max(5,0.24*sqrt(n)+2),L(n,q)*Psi(n,q)))
%o A000041 (PARI) a(n)=numbpart(n)
%o A000041 (PARI) a(n)=if(n<0,0,polcoeff(sum(k=1,sqrtint(n),x^k^2/prod(i=1,k,1-x^i,
1+x*O(x^n))^2,1),n))
%o A000041 (PARI) f(n)= {local(v,i,k,s,t);v=vector(n,k,0);v[n]=2;t=0;while(v[1]<n,
i=2;while(v[i]==0,i++);v[i]--;s=sum(k=i,n,k*v[k]); while(i>1,i--;
s+=i*(v[i]=(n-s)\i));t++);t } (Thomas Baruchel (baruchel(AT)users(AT)sourceforge.net),
Nov 07 2005)
%o A000041 (Mupad) combinat::partitions::count(i) $i=0..54 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 16 2007
%o A000041 (Other) sage: [number_of_partitions(n)for n in xrange(0,46)] # [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 24 2009]
%Y A000041 Cf. A000009, A008284, A008284, A000203, A001318.
%Y A000041 For successive differences see A002865, A053445, A072380, A081094, A081095.
%Y A000041 Antidiagonal sums of triangle A092905.
%Y A000041 Cf. A132311.
%Y A000041 Cf. A138151.
%Y A000041 Cf. A135010, A138121. [From Omar E. Pol (info(AT)polprimos.com), Sep
06 2008]
%Y A000041 A145006, A145007 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 28
2008]
%Y A000041 A080995 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 05 2008]
%Y A000041 A147843 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2008]
%Y A000041 Cf. A152537, A000079 [From Gary W. Adamson (qntmpkt(AT)haoo.com), Dec
06 2008]
%Y A000041 Sequence in context: A008641 A046054 A092885 this_sequence A084251 A024794
A091955
%Y A000041 a(n) = A035363(2n). [From Omar E. Pol (info(AT)polprimos.com), Nov 20
2009]
%Y A000041 Adjacent sequences: A000038 A000039 A000040 this_sequence A000042 A000043
A000044
%K A000041 core,easy,nonn,nice,new
%O A000041 0,3
%A A000041 N. J. A. Sloane (njas(AT)research.att.com).
%E A000041 Links contributed by Patrick De Geest (pdg(AT)worldofnumbers.com), Oct
15 1999. Additional comments from Ola Veshta (olaveshta(AT)my-deja.com),
Feb 28 2001 and from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com),
Apr 07 2001.
%E A000041 Further links contributed by Lekraj Beedassy (blekraj(AT)yahoo.com),
Spring 2003
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