%I A001147 M3002 N1217
%S A001147 1,1,3,15,105,945,10395,135135,2027025,34459425,654729075,13749310575,
%T A001147 316234143225,7905853580625,213458046676875,6190283353629375,191898783962510625,
%U A001147 6332659870762850625,221643095476699771875,8200794532637891559375
%N A001147 Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).
%C A001147 The solution to Schroeder's third problem.
%C A001147 a(n+2) is the number of full Steiner topologies on n points with n-2
Steiner points.
%C A001147 a(n) is also the number of perfect matchings in the complete graph K(2n)
- Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
%C A001147 Number of ways to choose n disjoint pairs of items from 2*n items. -
Ron Zeno (rzeno(AT)hotmail.com), Feb 06 2002
%C A001147 Also rational part of numerator of Gamma(n+1/2). Multiplying this sequence
by sqrt(Pi) and dividing by 2^n gives the value of Gamma(n+1/2).
- Yuriy Brun, Ewa Dominowska (brun(AT)mit.edu), May 12 2001
%C A001147 For n >= 1 a(n) is the number of permutations in the symmetric group
S_(2n) whose cycle decomposition is a product of n disjoint transpositions.
- Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001
%C A001147 Number of fixed-point-free involutions in symmetric group S_{2n}.
%C A001147 a(n) is the number of distinct products of n+1 variables with commutative,
nonassociative multiplication. - Andrew Walters (awalters3(AT)yahoo.com),
Jan 17 2004. For example, a(3)=15 because the product of the four
variables w, x, y and z can be constructed in exactly 15 ways, assuming
commutativity but not associativity: 1. w(x(yz)) 2. w(y(xz)) 3. w(z(xy))
4. x(w(yz)) 5. x(y(wz)) 6. x(z(wy)) 7. y(w(xz)) 8. y(x(wz)) 9. y(z(wx))
10. z(w(xy)) 11. z(x(wy)) 12. z(y(wx)) 13. (wx)(yz) 14. (wy)(xz)
15. (wz)(xy)
%C A001147 a(n) = E(X^(2n)), where X is a standard normal random variable (i.e.
X is normal with mean = 0, variance = 1). So for instance a(3) =
E(X^6) = 15, etc. See Abramowitz and Stegun or Hoel, Port and Stone.
- Jerome Coleman, Apr 06 2004
%C A001147 Second Eulerian transform of 1,1,1,1,1,1... The second Eulerian transform
transforms a sequence s to a sequence t by the formula t(n) = Sum[E(n,
k)s(k), k=0...n], where E(n,k) is a second-order Eulerian number
[A008517]. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 13 2005
%C A001147 Integral representation as n-th moment of a positive function on the
positive axis, in Maple notation: a(n)=int(x^n*exp(-x/2)/sqrt(2*Pi*x),
x=0..infinity), n=0,1... . - Karol A. Penson (penson(AT)lptl.jussieu.fr),
Oct 10 2005.
%C A001147 Let PI be the set of all partitions of {1, 2, ..., 2n} into pairs without
regard to order. There are (2n-1)!! such partitions. An element alpha
in PI can be written as alpha = {(i_1, j_1), (i_2, j_2), ..., (i_n,
j_n)} with i_k < j_k. Let pi be the corresponding permutation which
maps 1 to i_1, 2 to j_1, 3 to i_2, 4 to j_2, ..., 2n to j_n. Define
sgn(alpha) to be the signature of pi, which depends only on the partition
alpha and not on the particular choice of pi. Let A = {a_ij} be a
2n x 2n skew-symmetric matrix. Given a partition alpha as above define
A_alpha = sgn(alpha) a_(i_1,j_1)a_(i_2,j_2)...a_(i_n,j_n). We can
then define the Pfaffian of A to be Pf(A) = SUM[alpha in PI]A_alpha.
The Pfaffian of an n x n skew-symmetric matrix for n odd is defined
to be zero. - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 12
2006
%C A001147 a(n) is the number of binary total partitions (each non-singleton block
must be partitioned into exactly two blocks) or, equivalently, the
number of unordered full binary trees with labeled leaves (Stanley,
ex 5.2.6) - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Aug
01 2006
%C A001147 a(n) is the Pfaffian of the skew-symmetric 2n X 2n matrix whose (i,j)
entry is i for i<j. - David Callan (callan(AT)stat.wisc.edu), Sep
25 2006
%C A001147 a(n) is the number of increasing ordered rooted trees on n+1 vertices
where "increasing" means the vertices are labeled 0,1,2,...,n so
that each path from the root has increasing labels. Increasing unordered
rooted trees are counted by the factorial numbers A000142. - David
Callan (callan(AT)stat.wisc.edu), Oct 26 2006
%C A001147 Number of perfect multi Skolem-type sequences of order n. - Emeric Deutsch
(deutsch(AT)duke.poly.edu), Nov 24 2006
%C A001147 a(n) = total weight of all Dyck n-paths (A000108) when each path is weighted
with the product of the heights of the terminal points of its upsteps.
For example with n=3, the 5 Dyck 3-paths UUUDDD, UUDUDD, UUDDUD,
UDUUDD, UDUDUD have weights 1*2*3=6, 1*2*2=4, 1*2*1=2, 1*1*2=2, 1*1*1=1
respectively and 6+4+2+2+1=15. Counting weights by height of last
upstep yields A102625. - David Callan (callan(AT)stat.wisc.edu),
Dec 29 2006
%C A001147 a(n) is the number of increasing ternary trees on n vertices. Increasing
binary trees are counted by ordinary factorials (A000142) and increasing
quaternary trees by triple factorials (A007559). - David Callan (callan(AT)stat.wisc.edu),
Mar 30 2007
%C A001147 This sequence is essentially self-reciprocal under the list partition
transform and associated operations in A133314. More precisely, A001147
and -A001147 with a leading 1 attached are reciprocal. Therefore
their e.g.f.'s are reciprocal. See A132382 for an extension of this
result. - Tom Copeland (tcjpn(AT)msn.com), Nov 13 2007
%C A001147 Comments from Ross Drewe (rd(AT)labyrinth.net.au), Mar 16 2008: (Start)
This is also the number of ways of arranging the elements of n distinct
pairs, assuming the order of elements is significant but the pairs
are not distinguishable, i.e. arrangements which are the same after
permutations of the labels are equivalent.
%C A001147 If this sequence and A000680 are denoted by a(n) and b(n) respectively,
then a(n) = b(n)/n! where n! = the number of ways of permuting the
pair labels.
%C A001147 For example, there are 90 ways of arranging the elements of 3 pairs [1
1], [2 2], [3 3] when the pairs are distinguishable: A = { [112233],
[112323], .... [332211] }.
%C A001147 By applying the 6 relabeling permutations to A, we can partition A into
90/6 = 15 subsets: B = { {[112233], [113322], [221133], [223311],
[331122], [332211]}, {[112323], [113232], [221313], [223131], [331212],
[332121]}, ....}
%C A001147 Each subset or equivalence class in B represents a unique pattern of
pair relationships. For example, subset B1 above represents {3 disjoint
pairs} and subset B2 represents {1 disjoint pair + 2 interleaved
pairs}, with the order being significant (contrast A132101). (End)
%C A001147 A139541(n) = a(n) * a(2*n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Apr 25 2008
%C A001147 a(n+1) = sum(j=0 to n) A074060(n,j) * 2^j [From Tom Copeland (tcjpn(AT)msn.com),
Sep 01 2008]
%C A001147 Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 05 2009:
(Start)
%C A001147 a(n)=number of adjacent transpositions in all fixed-point-free involutions
of {1,2,...,2n}. Example: a(2)=3 because in 2143=(12)(34), 3412=(13)(24),
and 4321=(14)(23) we have 2 + 0 + 1 adjacent transpositions.
%C A001147 a(n)=Sum(k*A079267(n,k), k>=0).
%C A001147 (End)
%C A001147 Hankel transform is A137592. [From Paul Barry (pbarry(AT)wit.ie), Sep
18 2009]
%C A001147 (1, 3, 15, 105,...) = INVERT transform of A000698 starting (1, 2, 10,
74,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 21 2009]
%C A001147 a(n)=(-1)^(n+1)*H(2*n,0), where H(n,x) is the probabilists' Hermite polynomials.
The generating function for the the probabilists' Hermite polynomials
is as follows: exp(x*t-t^2/2)=sum(H(i,x)*t^i/i!,i=0,1,...) [From
Leonid Bedratyuk (leonid.uk(AT)gmail.com), Oct 31 2009]
%D A001147 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001147 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001147 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, Tenth Printing,
1972, (26.2.28).
%D A001147 D. Arques and J.-F. Beraud, Rooted maps on orientable surfaces..., Discrete
Math., 215 (2000), 1-12.
%D A001147 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 228, #19.
%D A001147 Thierry Dana-Picard, Sequences of Definite Integrals, Factorials and
Double Factorials, Journal of Integer Sequences, Vol. 8 (2005), Article
05.4.6.
%D A001147 Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham,
Eulerian, MacMahon and Stirling number triangles, Journal of Integer
Sequences, Vol. 9 (2006), Article 06.4.1.
%D A001147 Hoel, Port and Stone, Introduction to Probability Theory, Section 7.3.
%D A001147 F. K. Hwang, D. S. Richards and P. Winter, The Steiner Tree Problem,
North-Holland, 1992, see p. 14.
%D A001147 L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p.
48
%D A001147 M. Klazar, Twelve countings with rooted plane trees, European Journal
of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
%D A001147 B. E. Meserve, Double factorials, Amer. Math. Monthly, 55 (1948), 425-426.
%D A001147 T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51
(1945), 976-984.
%D A001147 F. Murtagh, "Counting dendrograms: a survey", Discrete Applied Mathematics,
7 (1984), 191-199.
%D A001147 R. Ondrejka, Tables of double factorials, Math. Comp., 24 (1970), 231.
%D A001147 E. Schroeder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870),
361-376.
%D A001147 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see
Example 5.2.6 and also p. 178.
%H A001147 T. D. Noe, <a href="b001147.txt">Table of n, a(n) for n=0..101</a>
%H A001147 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 A001147 H. Bottomley, <a href="a002694.gif">Illustration for A000108, A001147,
A002694, A067310 and A067311</a>
%H A001147 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 A001147 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=23">
Encyclopedia of Combinatorial Structures 23</a>
%H A001147 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=106">
Encyclopedia of Combinatorial Structures 106</a>
%H A001147 A. Khruzin, <a href="http://arXiv.org/abs/math.CO/0008209">Enumeration
of chord diagrams</a>
%H A001147 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
On generalizations of Stirling number triangles</a>, J. Integer Seqs.,
Vol. 3 (2000), #00.2.4.
%H A001147 G. Nordh, <a href="http://arXiv.org/abs/math.NT/0506155">Perfect Skolem
sequences</a>
%H A001147 L. Pachter and B. Sturmfels, <a href="http://arXiv.org/abs/math.ST/0409132">
The mathematics of phylogenomics</a>
%H A001147 Helmut Prodinger, <a href="http://www.combinatorics.org/Volume_3/volume3.html#R29">
Descendants in heap ordered trees or a triumph of computer algebra
</a>
%H A001147 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
question/q541.htm">Question 541</a>, J. Ind. Math. Soc.
%H A001147 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
DoubleFactorial.html">Link to a section of The World of Mathematics.</
a>
%H A001147 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
GammaFunction.html">Link to a section of The World of Mathematics.</
a>
%H A001147 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
NormalDistributionFunction.html">Link to a section of The World of
Mathematics.</a>
%H A001147 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Erf.html">Erf</a>
%H A001147 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pfaffian">Pfaffian</
a>
%H A001147 <a href="Sindx_Par.html#partN">Index entries for related partition-counting
sequences</a>
%H A001147 <a href="Sindx_Fa.html#factorial">Index entries for sequences related
to factorial numbers</a>
%H A001147 <a href="Sindx_Par.html#parens">Index entries for sequences related to
parenthesizing</a>
%H A001147 <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite_polynomials</
a> [From Leonid Bedratyuk (leonid.uk(AT)gmail.com), Oct 31 2009]
%F A001147 E.g.f.: 1/sqrt(1-2x). a(n) = a(n-1)*(2n-1) = (2n)!/(n!*2^n) = A010050(n)/
A000165(n). a(n) ~ sqrt(2) * 2^n * (n/e)^n.
%F A001147 With interpolated zeros, the sequence has e.g.f. exp(x^2/2). - Paul Barry
(pbarry(AT)wit.ie), Jun 27 2003
%F A001147 The Ramanujan polynomial psi(n+1, n) has value a(n). - R. Stephan, Apr
16 2004
%F A001147 a(n) = Sum_{k=0..n} (-2)^(n-k)*A048994(n, k) .- Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Oct 29 2005
%F A001147 log(1+x+3*x^2+15*x^3+105*x^4+945*x^5+10395*x^6+...)=x+5/2*x^2+37/3*x^3+353/
4*x^4+4081/5*x^5+55205/6*x^6+..., where [1, 5, 37, 353, 4081, 55205,
...] = A004208 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun
20 2006
%F A001147 1/3 + 2/15 + 3/105 +...= 1/2 1/1 + 1/3 + 2/15 + 6/105 + 24/945 +...=
Pi/2 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2006
%F A001147 a(n)=(1/sqrt(2*pi))*int(x^n*exp(-x/2)/sqrt(x),x,0,infty); - Paul Barry
(pbarry(AT)wit.ie), Jan 28 2008
%F A001147 a(n)=A006882(2n-1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Jul 04 2009]
%F A001147 G.f.: 1/(1-x-2x^2/(1-5x-12x^2/(1-9x-30x^2/(1-13x-56x^2/(1- ... (continued
fraction). [From Paul Barry (pbarry(AT)wit.ie), Sep 18 2009]
%F A001147 a(n)=(-1)^n*subs({ln(e)=1,x=0},coeff(simplify(series(e^(x*t-t^2/2),t,
2*n+1)),t^(2*n))*(2*n)!) [From Leonid Bedratyuk (leonid.uk(AT)gmail.com),
Oct 31 2009]
%F A001147 a(n)=2^n*gamma(n+1/2)/gamma(1/2) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com),
Nov 09 2009]
%e A001147 a(3)=1*3*5=15.
%p A001147 f := n->(2*n)!/(n!*2^n);
%p A001147 A001147 := proc(n) doublefactorial(2*n-1); end: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Jul 04 2009]
%p A001147 A001147 := n -> 2^n*pochhammer(1/2, n); (From Peter Luschny, Aug 09 2009)
%p A001147 with(finance):seq(mul(cashflows([k,k,1],0), k=0..n-1), n=0..19);# [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 22 2008]
%p A001147 restart: G(x):=(1-2*x)^(-1/2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],
x) od: x:=0: seq(f[n],n=0..19);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 03 2009]
%t A001147 Table[(2n - 1)!!, {n, 0, 19}] (from Robert G. Wilson v (rgwv(at)rgwv.com),
Oct 12 2005)
%o A001147 (PARI) a(n)=if(n<0,0,(2*n)!/n!/2^n)
%Y A001147 Cf. A006882, A076795, A000165 ((2n)!!), A001818, A009445, A039683. a(n)=
A035342(n, 1), n >= 1 (first column of triangle).
%Y A001147 Cf. A086677; A055142 (for this sequence, |a(n+1)| + 1 is the number of
distinct products which can be formed using commutative, nonassociative
multiplication and a nonempty subset of n given variables).
%Y A001147 Constant terms of polynomials in A098503. First row of array A099020.
%Y A001147 Cf. A102992, A001190 (no labels).
%Y A001147 a(n)=A001497(n, 0) = A001498(n, n), first column, resp. main diagonal,
of Bessel triangle.
%Y A001147 Cf. A000680, A132101.
%Y A001147 A079267 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 05 2009]
%Y A001147 Cf. A000698 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 21 2009]
%Y A001147 Sequence in context: A001801 A067546 A015682 this_sequence A000268 A118750
A070826
%Y A001147 Adjacent sequences: A001144 A001145 A001146 this_sequence A001148 A001149
A001150
%K A001147 nonn,easy,nice,core,new
%O A001147 0,3
%A A001147 N. J. A. Sloane (njas(AT)research.att.com).
%E A001147 More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Apr 25 2008
%E A001147 Removed erroneous entries; neither the number of n x n binary matrices
A such that A^2 = 0 nor the number of simple directed graphs on n
vertices with no directed path of length two are counted by this
sequence (for n = 3, both are 13) Dan Drake (ddrake(AT)member.ams.org),
Jun 02 2009
%E A001147 Maple program Zerinvary Lajos aligned with offset Johannes W. Meijer
(meijgia(AT)hotmail.com), Aug 11 2009
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