%I A000984 M1645 N0643
%S A000984 1,2,6,20,70,252,924,3432,12870,48620,184756,705432,2704156,10400600,
%T A000984 40116600,155117520,601080390,2333606220,9075135300,35345263800,
%U A000984 137846528820,538257874440,2104098963720,8233430727600,32247603683100
%N A000984 Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2.
%C A000984 Equal to the binomial coefficient sum Sum_{k=0..n} binomial(n,k)^2.
%C A000984 Number of possible interleavings of a program with n atomic instructions
when executed by two processes - Manuel Carro (mcarro(AT)fi.upm.es),
Sep 22 2001
%C A000984 Convolving a(n) with itself yields A000302, the powers of 4. - T. D.
Noe (noe(AT)sspectra.com), Jun 11 2002
%C A000984 a(n)=Max{ (i+j)!/(i!j!) | 0<=i,j<=n } - Benoit Cloitre (benoit7848c(AT)orange.fr),
May 30 2002
%C A000984 Number of ordered trees with 2n+1 edges, having root of odd degree and
nonroot nodes of outdegree 0 or 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Aug 02 2002
%C A000984 Also number of directed, convex polyominoes having semiperimeter n+2.
%C A000984 Also number of diagonally symmetric, directed, convex polyominoes having
semiperimeter 2n+2. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Aug 03 2002
%C A000984 Also Sum_{k=0..n} binomial(n+k-1,k). - Vladeta Jovovic (vladeta(AT)eunet.rs),
Aug 28 2002
%C A000984 The second inverse binomial transform of this sequence is this sequence
with interpolated zeros. Its G.f. is (1 - 4*x^2)^(-1/2), with n-th
term C(n,n/2)(1+(-1)^n)/2. - Paul Barry (pbarry(AT)wit.ie), Jul 01
2003
%C A000984 Number of possible values of a 2*n bit binary number for which half the
bits are on and half are off. - Gavin Scott (gavin(AT)allegro.com),
Aug 09 2003
%C A000984 Ordered partitions of n with zeros to n+1, e.g. for n=4 we consider the
ordered partitions of 11110 (5), 11200 (30), 13000 (20), 40000 (5)
and 22000 (10), total 70 and a(4)=70. See A001700 (esp. Mambetov
Bektur's comment). - Jon Perry (perry(AT)globalnet.co.uk), Aug 10
2003
%C A000984 Number of non-decreasing sequences of n integers from 0 to n: a(n) =
sum_{i_{1}=0}^{n} sum_{i_{2}=i_{1}}^{n}...sum_{i_{n}=i_{n-1}}^{n}(1).
- J. N. Bearden (jnb(AT)eller.arizona.edu), Sep 16 2003
%C A000984 Number of peaks at odd level in all Dyck paths of semilength n+1. Example:
a(2)=6 because we have U*DU*DU*D, U*DUUDD, UUDDU*D, UUDUDD, UUU*DDD,
where U=(1,1), D=(1,-1) and * indicates a peak at odd level. Number
of ascents of length 1 in all Dyck paths of semilength n+1 (an ascent
in a Dyck path is a maximal string of up steps). Example: a(2)=6
because we have uDuDuD, uDUUDD, UUDDuD, UUDuDD, UUUDDD, where an
ascent of length 1 is indicated by a lower case letter. - Emeric
Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003
%C A000984 a(n-1)=number of subsets of 2n-1 distinct elements taken n at a time
that contain a given element. e.g. n=4 -> a(3)=20 and if we consider
the subsets of 7 taken 4 at a time with a 1 we get (1234, 1235, 1236,
1237, 1245, 1246, 1247, 1256, 1257, 1267, 1345, 1346, 1347, 1356,
1357, 1367, 1456, 1457, 1467, 1567) and there are 20 of them. - Jon
Perry (perry(AT)globalnet.co.uk), Jan 20 2004
%C A000984 The dimension of a particular (necessarily existent) absolutely universal
embedding of the unitary dual polar space DSU(2n,q^2) where q>2.
- J. Taylor (jt_cpp(AT)yahoo.com), Apr 02 2004.
%C A000984 Number of standard tableaux of shape (n+1, 1^n). - Emeric Deutsch (deutsch(AT)duke.poly.edu),
May 13 2004
%C A000984 Erdos, Graham et al. conjectured that a(n) is never squarefree for sufficiently
large n. Sarkozy showed that if s(n) is the square part of a(n),
then s(n) is asymptotically (sqrt(2)-2)*(sqrt(n))*(Riemann Zeta Function(1/
2)). Granville and Ramare proved that the only squarefree values
are a(1)=2, a(2)=6 and a(4)=70. A000984(n)/(n+1) = A000108(n), that
is, dividing by (n+1) scales the Central binomial coefficients to
Catalan numbers also called Segner numbers. - Jonathan Vos Post (jvospost3(AT)gmail.com),
Dec 04 2004
%C A000984 p divides a((p-1)/2)-1=A030662[n] for prime p=5,13,17,29,37,41,53,61,
73,89,97..=A002144[n] Pythagorean primes: primes of form 4n+1. -
Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 04 2006
%C A000984 The number of direct routes from my home to Granny's when Granny lives
n blocks south and n blocks east of my home in Grid City. To obtain
a direct route, from the 2n blocks, choose n blocks on which one
travels south. For example, a(2)=6 because there are 6 direct routes:
SSEE, SESE, SEES, EESS, ESES and ESSE. - Dennis P. Walsh (dwalsh(AT)mtsu.edu),
Oct 27 2006
%C A000984 Inverse: With q = -log(log(16)/(pi a(n)^2)), ceiling((q + log(q))/log(16))
= n. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 26 2007
%C A000984 Number of partitions with Ferrers diagrams that fit in an n X n box (including
the empty partition of 0). Example: a(2) = 6 because we have: empty,
1, 2, 11, 21 and 22. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Oct 02 2007
%C A000984 The number of walks of length 2n on an infinite linear lattice that begin
and end at the origin. - Stefan Hollos (stefan(AT)exstrom.com), Dec
10 2007
%C A000984 Integral representation : C(2n,n)=1/Pi Integral [(2x)^(2n)/Sqrt[1 - x^2],
{x,-1, 1}], i.e. C(2n,n)/4^n is the moment of order 2n of the arcsin
distribution on the interval (-1,1). - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr),
Jan 02 2008
%C A000984 Define the array m(1,j)=1 ; m(i,1)=1 ; m(i,j)=m(i,j-1) + m(j,i-1), then
a(n) = m(n,n) [From philippe lallouet (philip.lallouet(AT)orange.fr),
Sep 15 2008]
%C A000984 Also the Catalan transform of A000079. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Nov 06 2008]
%C A000984 Straub, Amdeberhan and Moll: "... it is conjectured that there are only
finitely many indices n such that C_n is not divisible by any of
3, 5, 7 and 11. Finally, we remark that Erdos et al. conjectured
that the central binomial coefficients C_n are never squarefree for
n > 4 which has been proved by Granville and Ramare." [From Jonathan
Vos Post (jvospost3(AT)gmail.com), Nov 14 2008]
%C A000984 Equals row sums of triangle A152229 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Nov 29 2008]
%C A000984 Equals row sums of triangle A158815 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Mar 27 2009]
%C A000984 This sequence appears in formulae in the link cited. [Oktay Haracci (timetunnel3(AT)hotmail.com),
Apr 02 2009]
%C A000984 Equals INVERT transform of A081696: (1, 1, 3, 9, 29, 97, 333,...). [From
Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]
%C A000984 Also, in sports, the number of ordered ways for a "Best of 2n-1 Series"
to progress. For example, a(2) = 6 means there are six ordered ways
for a "best of 3" series to progress. If we write A for a win by
"team A" and B for a win by "team B" and if we list the played games
chronologically from left to right then the six ways are AA, ABA,
BAA, BB, BAB, and ABB. (Proof: To generate the a(n) ordered ways:
Write down all a(n) ways to designate n of 2n games as won by team
A. Remove the maximal suffix of identical letters from each of these.)
[From Lee A. Newberg (integer(AT)quantconsulting.com), Jun 02 2009]
%C A000984 Contribution from Jason Richardson-White (coyoteworks(AT)gmail.com),
Jun 15 2009: Index the central binomial coefficients with the natural
numbers 1,2,3...,n. It appears that dividing the central binomial
coefficients by their indexes yields the Catalan numbers (A000108).
%C A000984 Number of nXn binary arrays with rows, considered as binary numbers,
in nondecreasing order, and columns, considered as binary numbers,
in nonincreasing order. [From Ron Hardin (rhhardin(AT)att.net), Jun
27 2009]
%C A000984 Hankel transform is 2^n. [From Paul Barry (pbarry(AT)wit.ie), Aug 05
2009]
%D A000984 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000984 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000984 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, 1964 (and various
reprintings), p. 828.
%D A000984 M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008),
2544-2563.
%D A000984 Paul Barry, A Catalan Transform and Related Transformations on Integer
Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A000984 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal
Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%D A000984 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of
combinatorial proof, M.A.A. 2003, id. 160.
%D A000984 A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining
convex permutominoes, preprint, 2007.
%D A000984 Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer
Sequences, Vol. 9 (2006), Article 06.2.3.
%D A000984 Thierry Dana-Picard, Sequences of Definite Integrals, Factorials and
Double Factorials, Journal of Integer Sequences, Vol. 8 (2005), Article
05.4.6.
%D A000984 E. Deutsch and L. Shapiro, Seventeen Catalan identities, Bulletin of
the Institute of Combinatorics and its Applications, 31, 31-38, 2001.
%D A000984 Erdos, P.; Graham, R. L.; Ruzsa, I. Z.; and Straus, E. G. "On the Prime
Factors of C(2n,n)." Math. Comput. 29, 83-92, 1975.
%D A000984 H. W. Gould, Combinatorial Identities, Morgantown, 1972, (3.66), page
30.
%D A000984 M. Griffiths, The Backbone of Pascal's Triangle, United Kingdom Mathematics
Trust (2008), 3-124. [From Martin Griffiths (griffm(AT)essex.ac.uk),
Mar 28 2009]
%D A000984 Granville, A. and Ramare, O. "Explicit Bounds on Exponential Sums and
the Scarcity of Squarefree Binomial Coefficients." Mathematika 43,
73-107, 1996.
%D A000984 J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society
Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D A000984 T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51
(1945), 976-984.
%D A000984 Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients,
Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%D A000984 Sarkozy, A. "On Divisors of Binomial Coefficients. I." J. Number Th.
20, 70-80, 1985.
%D A000984 L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group,
Discrete Applied Math., 34 (1991), 229-239.
%D A000984 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the
Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article
06.1.1.
%H A000984 T. D. Noe, <a href="b000984.txt">Table of n, a(n) for n = 0..200</a>
%H A000984 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 A000984 D. H. Bailey, J. M. Borwein and D. M. Bradley, <a href="http://arXiv.org/
abs/math.CA/0505270">Experimental determination of Ap'ery-like identities
for zeta(4n+2)</a>
%H A000984 J. Borwein and D. Bradley, <a href="http://arXiv.org/abs/math.CA/0505124">
Empirically determined Ap'ery-like formulae for zeta(4n+3)</a>
%H A000984 N. T. Cameron, <a href="http://www.princeton.edu/~wmassey/NAM03/cameron.pdf">
Random walks, trees and extensions of Riordan group techniques</a>
%H A000984 B. N. Cooperstein and E. E. Shult, <a href="http://www.emis.de/journals/
AG/1-1/1_037.pdf">A note on embedding and generating dual polar spaces</
a>. Adv. Geom. 1 (2001), 37-48. See Theorem 5.4.
%H A000984 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/manhattan.html">
Shortest-path diagrams</a>
%H A000984 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
Publications/books.html">Analytic Combinatorics</a>, 2009; see page
77
%H A000984 Oktay Haracci (timetunnel3(AT)hotmail.com), <a href="http://www.geocities.com/
timeparadox/ismi_azam.html">Regular Polygons</a>
%H A000984 Ron Hardin, <a href="a151801.txt">Binary arrays with both rows and cols
sorted, symmetries</a>
%H A000984 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A000984 I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/polygons/Polygons_ser.html">
Series exapansions for self-avoiding polygons</a>
%H A000984 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Matrix Transformations of Integer Sequences</a>, J. Integer Seqs.,
Vol. 6, 2003.
%H A000984 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
The Hankel Transform and Some of its Properties</a>, J. Integer Sequences,
4 (2001), #01.1.5.
%H A000984 L. Lipshitz and A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/
alfpapers/a084.pdf">Rational functions, diagonals, automata and arithmetic</
a>
%H A000984 P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/
JIS/index.html">Generating Functions via Hankel and Stieltjes Matrices</
a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
%H A000984 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">Arithmetic and growth of periodic orbits</a>, J. Integer
Seqs., Vol. 4 (2001), #01.2.1.
%H A000984 Armin Straub, Tewodros Amdeberhan and Victor H. Moll, <a href="http:/
/arxiv.org/abs/0811.2028">The p-adic valuation of k-central binomial
coefficients</a>, Nov 13, 2008, pp. 10-11. [From Jonathan Vos Post
(jvospost3(AT)gmail.com), Nov 14 2008]
%H A000984 V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">
Recurrences and Legendre transform</a>
%H A000984 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol.
3 (2000), #00.1.
%H A000984 H. A. Verrill, <a href="http://arXiv.org/abs/math.CO/0407327">Sums of
squares of binomial coefficients, ...</a>
%H A000984 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
BinomialSums.html">Link to a section of The World of Mathematics.</
a>
%H A000984 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CentralBinomialCoefficient.html">Link to a section of The World of
Mathematics.</a>
%H A000984 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
StaircaseWalk.html">Link to a section of The World of Mathematics.</
a>
%H A000984 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CircleLinePicking.html">Circle Line Picking</a>
%H A000984 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000984 G.f.: A(x) = (1 - 4*x)^(-1/2) = 1 + 2*x + 6*x^2 + 20*x^3 + ...
%F A000984 a(n) = 2^n/n! * product[ k=0..n-1 ] (2*k+1).
%F A000984 a(n) = a(n-1)*(4-2/n) = 4a(n-1)+A002420(n) = A000142(2n)/(A000142(n)^2)
= A001813(n)/A000142(n) = sqrt(A002894(n)) = A010050(n)/A001044(n)
= (n+1)*A000108(n) = -A005408(n-1)*A002420(n) - Henry Bottomley (se16(AT)btinternet.com),
Nov 10 2000
%F A000984 Using Stirling's formula in A000142 it is easy to get the asymptotic
expression a(n) ~ 4^n / sqrt(Pi * n) - Dan Fux (dan.fux(AT)OpenGaia.com
or danfux(AT)OpenGaia.com), Apr 07 2001
%F A000984 Integral representation as n-th moment of a positive function on the
interval[0, 4], in Maple notation: a(n)= int(x^n*((x*(4-x))^(-1/2))/
Pi, x=0..4), n=0, 1, ... This representation is unique. - Karol A.
Penson (penson(AT)lptl.jussieu.fr), Sep 17 2001
%F A000984 sum(n>=1, 1/a(n))=(2*Pi*sqrt(3)+9)/27 - Benoit Cloitre (benoit7848c(AT)orange.fr),
May 01 2002
%F A000984 E.g.f.: exp(2x) I_0(2x), where I_0 is Bessel function. - Michael Somos,
Sep 08 2002
%F A000984 E.g.f.: I_0(2x)=sum a(n) x^(2n)/(2n)!, where I_0 is Bessel function.
- Michael Somos, Sep 09, 2002.
%F A000984 a(n) = sum(k=0, n, C(n, k)^2). - Benoit Cloitre (benoit7848c(AT)orange.fr),
Jan 31 2003
%F A000984 Determinant of n X n matrix M(i, j)=binomial(n+i, j) - Benoit Cloitre
(benoit7848c(AT)orange.fr), Aug 28 2003
%F A000984 Given m = C(2n, n), let f be the inverse function, so that f(m) = n.
Letting q denote -Log(Log(16)/(m^2*Pi)), we have f(m) = Ceiling(
(q + Log(q)) / Log(16) ). - David W. Cantrell (DWCantrell(AT)sigmaxi.net),
Oct 30 2003
%F A000984 a(n) = 2*Sum{k= 0...(n-1), a(k)*a(n-k+1)/(k+1)}. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Jan 01 2004
%F A000984 a(n+1)=sum(j=n, n*2+1, binomial(j, n)). E.g. a(4)=C(7, 3)+C(6, 3)+C(5,
3)+C(4, 3)+C(3, 3)=35+20+10+4+1=70 - Jon Perry (perry(AT)globalnet.co.uk),
Jan 20 2004
%F A000984 a(n) = (-1)^(n)*sum(j=0..(2*n), (-1)^j*binomial(2*n, j)^2) - Helena Verrill
(verrill(AT)math.lsu.edu), Jul 12 2004
%F A000984 a(n)=sum{k=0..n, binomial(2n+1, k)*sin((2n-2k+1)*pi/2)}. - Paul Barry
(pbarry(AT)wit.ie), Nov 02 2004
%F A000984 a(n-1)=(1/2)*(-1)^n*sum_{0<=i, j<=n}(-1)^(i+j)*binomial(2n, i+j) - Benoit
Cloitre (benoit7848c(AT)orange.fr), Jun 18 2005
%F A000984 a(n) = C(2n, n-1) + C(n) = A001791(n) + A000108(n). a(n) = (n+1)*C(n)
= (n+1)*A000108(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Aug
02 2005
%F A000984 G.f.: c(x)^2/(2*c(x)-c(x)^2) where c(x) is the g.f. of A000108; - Paul
Barry (pbarry(AT)wit.ie), Feb 03 2006
%F A000984 a(n)=A006480(n)/A005809(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 28 2007
%F A000984 a(n)=Sum{k, 0<=k<=n}A106566(n,k)*2^k. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Aug 25 2007
%F A000984 a(n)= Sum{k>=0, A039599(n, k)} . a(n)= Sum{k>=0, A050165(n, k)} . a(n)=
Sum{k>=0, A059365(n, k)*2^k}, n>0 . a(n+1)= Sum{k>=0, A009766(n,
k)*2^(n-k+1)}. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 01
2004
%F A000984 a(n)=4^n*sum{k=0..n, C(n,k)(-4)^(-k)*A000108(n+k)}; - Paul Barry (pbarry(AT)wit.ie),
Oct 18 2007
%F A000984 Row sums of triangle A135091 - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Nov 18 2007
%F A000984 a(n)=Sum_{k, 0<=k<=n}A039598(n,k)*A059841(k). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Nov 12 2008]
%F A000984 A007814(a(n))=A000120(n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il),
Jul 20 2009]
%F A000984 Contribution from Paul Barry (pbarry(AT)wit.ie), Aug 05 2009: (Start)
%F A000984 G.f.: 1/(1-2x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction);
%F A000984 G.f.: 1/(1-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)
%F A000984 a(n)=Product(k=1..n)[4-2/k] [From David Brown (thedabs(AT)gmail.com),
Sep 19 2009]
%p A000984 A000984 := n-> binomial(2*n,n);
%p A000984 with(combstruct); [seq(count([S,{S=Prod(Set(Z,card=i),Set(Z,card=i))},
labeled],size=(2*i)),i =0..20)];
%p A000984 with(combstruct); [seq(count([S,{S=Sequence(Union(Arch,Arch)), Arch=Prod(Epsilon,
Sequence(Arch),Z)},unlabeled],size=i), i=0..25)];
%p A000984 Z:=(1-sqrt(1-z))*4^n/sqrt(1-z): Zser:=series(Z, z=0, 32): seq(coeff(Zser,
z, n), n=0..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jan 01 2007
%p A000984 with(combstruct):bin := {B=Union(Z,Prod(B,B))}: seq (count([B,bin,unlabeled],
size=n)*n, n=1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Dec 05 2007
%t A000984 Table[Binomial[2n, n], {n, 0, 24}] (Alonso Delarte (alonso.delarte(AT)gmail.com),
Nov 10 2005)
%o A000984 (MAGMA) a:= func< n | Binomial(2*n,n) >; [ a(n) : n in [0..10]];
%o A000984 (PARI) a(n)=if(n<0,0,(2*n)!/n!^2)
%Y A000984 A000984(n+1)=2*A001700(n)=A030662(n)+1. a(2*n) = A001448(n), a(2*n+1)
= 2*A002458(n).
%Y A000984 Cf. A000108, A002420, A002457. Differs from A071976 at 10-th term.
%Y A000984 Bisection of A001405. Row sums of A059481.
%Y A000984 Row sums of triangle A008459.
%Y A000984 Cf. A030662, A002144.
%Y A000984 Cf. A135091.
%Y A000984 A152229 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]
%Y A000984 A158815 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 27 2009]
%Y A000984 A081696 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]
%Y A000984 Sequence in context: A056616 A065346 A071976 this_sequence A087433 A119373
A151284
%Y A000984 Adjacent sequences: A000981 A000982 A000983 this_sequence A000985 A000986
A000987
%K A000984 nonn,easy,core,nice
%O A000984 0,2
%A A000984 N. J. A. Sloane (njas(AT)research.att.com).
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