A000085 -id:A000085 - cp's OEIS frontend

A006882 Double factorials n!!: a(n) = n*a(n-2) for n > 1, a(0) = a(1) = 1.

1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425, 185794560, 654729075, 3715891200, 13749310575, 81749606400, 316234143225, 1961990553600, 7905853580625, 51011754393600, 213458046676875, 1428329123020800

Offset: 0

Robert Munafo

Product of pairs of successive terms gives factorials in increasing order. - Amarnath Murthy, Oct 17 2002
a(n) = number of down-up permutations on [n+1] for which the entries in the even positions are increasing. For example, a(3)=3 counts 2143, 3142, 4132. Also, a(n) = number of down-up permutations on [n+2] for which the entries in the odd positions are decreasing. For example, a(3)=3 counts 51423, 52413, 53412. - David Callan, Nov 29 2007
The double factorial of a positive integer n is the product of the positive integers <= n that have the same parity as n. - Peter Luschny, Jun 23 2011
For n even, a(n) is the number of ways to place n points on an n X n grid with pairwise distinct abscissa, pairwise distinct ordinate, and 180-degree rotational symmetry. For n odd, the number of ways is a(n-1) because the center point can be considered "fixed". For 90-degree rotational symmetry cf. A001813, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017
Could be extended to include a(-1) = 1. But a(-2) is not defined, otherwise we would have 1 = a(0) = 0*a(-2). - Jianing Song, Oct 23 2019

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 15*x^5 + 48*x^6 + 105*x^7 + 384*x^8 + ...

Putnam Contest, 4 Dec. 2004, Problem A3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Indranil Ghosh, Table of n, a(n) for n = 0..806 (terms 0..100 from T. D. Noe)
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.
CombOS - Combinatorial Object Server, Generate colored permutations
Joseph E. Cooper III, A recurrence for an expression involving double factorials, arXiv:1510.00399 [math.CO], 2015.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Gary T. Leavens and Mike Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
Peter Luschny, Multifactorials
B. E. Meserve, Double Factorials, American Mathematical Monthly, 55 (1948), 425-426.
Rudolph Ondrejka, Tables of double factorials, Math. Comp., Vol. 24, No. 109 (1970), p. 231.
Eric Weisstein's World of Mathematics, Double Factorial.
Eric Weisstein's World of Mathematics, Multifactorial.
Index entries for sequences related to factorial numbers
Index entries for "core" sequences

Bisections are A000165 and A001147. These two entries have more information.
Cf. A052319, A143280.
A diagonal of A202212.
Cf. A000085, A001813, A135401, A297708. - Manfred Scheucher, Jan 07 2018

a006882 n = a006882_list !! n
a006882_list = 1 : 1 : zipWith (*) [2..] a006882_list
-- Reinhard Zumkeller, Oct 23 2014

DoubleFactorial:=func< n | &*[n..2 by -2] >; [ DoubleFactorial(n): n in [0..28] ]; // Klaus Brockhaus, Jan 23 2011

A006882 := proc(n) option remember; if n <= 1 then 1 else n*A006882(n-2); fi; end;
A006882 := proc(n) doublefactorial(n) ; end proc; seq(A006882(n),n=0..10) ; # R. J. Mathar, Oct 20 2009
A006882 := n -> mul(k, k = select(k -> k mod 2 = n mod 2, [$1 .. n])):  seq(A006882(n), n = 0 .. 10); # Peter Luschny, Jun 23 2011
A006882 := proc(n) if n=0 then 1 else mul(n-2*k, k=0..floor(n/2)-1); fi; end; # N. J. A. Sloane, May 27 2016

Array[ #!!&, 40, 0 ]
multiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*multiFactorial[n - k, k]]]; Array[ multiFactorial[#, 2] &, 27, 0] (* Robert G. Wilson v, Apr 23 2011 *)

{a(n) = prod(i=0, (n-1)\2, n - 2*i )} \\ Improved by M. F. Hasler, Nov 30 2013

{a(n) = if( n<2, n>=0, n * a(n-2))}; /* Michael Somos, Apr 06 2003 */

{a(n) = if( n<0, 0, my(E); E = exp(x^2 / 2 + x * O(x^n)); n! * polcoeff( 1 + E * x * (1 + intformal(1 / E)), n))}; /* Michael Somos, Apr 06 2003 */

from sympy import factorial2
def A006882(n): return factorial2(n) # Chai Wah Wu, Apr 03 2021

a(n) = Product_{i=0..floor((n-1)/2)} (n - 2*i).
E.g.f.: 1+exp(x^2/2)*x*(1+sqrt(Pi/2)*erf(x/sqrt(2))). - Wouter Meeussen, Mar 08 2001
Satisfies a(n+3)*a(n) - a(n+1)*a(n+2) = (n+1)!. [Putnam Contest]
a(n) = n!/a(n-1). - Vaclav Kotesovec, Sep 17 2012
a(n) * a(n+3) = a(n+1) * (a(n+2) + a(n)). a(n) * a(n+1) = (n+1)!. - Michael Somos, Dec 29 2012
a(n) ~ c * n^((n+1)/2) / exp(n/2), where c = sqrt(Pi) if n is even, and c = sqrt(2) if n is odd. - Vaclav Kotesovec, Nov 08 2014
a(2*n) = 2^n*a(n)*a(n-1). a(2^n) = 2^(2^n - 1) * 1!! * 3!! * 7!! * ... * (2^(n-1) - 1)!!. - Peter Bala, Nov 01 2016
a(n) = 2^h*(2/Pi)^(sin(Pi*h)^2/2)*Gamma(h+1) where h = n/2. This analytical extension supports the view that a(-1) = 1 is a meaningful numerical extension. With this definition (-1/2)!! = Gamma(3/4)/Pi^(1/4). - Peter Luschny, Oct 24 2019
a(n) ~ (n+1/6)*sqrt((2/e)*(n/e)^(n-1)*(Pi/2)^(cos(n*Pi/2)^2)). - Peter Luschny, Oct 25 2019
Sum_{n>=0} 1/a(n) = A143280. - Amiram Eldar, Nov 10 2020
Sum_{n>=0} 1/(a(n)*a(n+1)) = e - 1. - Andrés Ventas, Apr 12 2021

A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

Original entry on oeis.org

1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000, 830034394580628357120000, 58102407620643984998400000

Offset: 0

N. J. A. Sloane

Counts binary rooted trees (with out-degree <= 2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.
Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003
Number of broadcast schemes in the complete graph on n+1 vertices, K_{n+1}. - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
Hankel transform is A137565. - Paul Barry, Nov 25 2009
The e.g.f. of 1/a(n) = n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. - Wolfdieter Lang, Jan 09 2012
From Tom Copeland, Nov 15 2014: (Start)
Aerated with intervening zeros (1,0,2,0,12,0,120,...) = a(n) (cf. A123023 and A001147), the e.g.f. is e^(t^2), so this is the base for the Appell sequence with e.g.f. e^(t^2) e^(x*t) = exp(P(.,x),t) (reverse A059344, cf. A099174, A066325 also). P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for e^(-t^2)e^(x*t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), e.g., (P(.,t))^n = P(n,t).
Equals A000407*2 with leading 1 added. (End)
a(n) is also the number of square roots of any permutation in S_{4*n} whose disjoint cycle decomposition consists of 2*n transpositions. - Luis Manuel Rivera Martínez, Mar 04 2015
Self-convolution gives A076729. - Vladimir Reshetnikov, Oct 11 2016
For n > 1, it follows from the formula dated Aug 07 2013 that a(n) is a Zumkeller number (A083207). - Ivan N. Ianakiev, Feb 28 2017
For n divisible by 4, a(n/4) is the number of ways to place n points on an n X n grid with pairwise distinct abscissae, pairwise distinct ordinates, and 90-degree rotational symmetry. For n == 1 (mod 4), the number of ways is a((n-1)/4) because the center point can be considered "fixed". For 180-degree rotational symmetry see A006882, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017

			The following permutations of order 8 and their reversals have this property:
  1 7 3 5 2 4 0 6
  1 7 4 2 5 3 0 6
  2 3 7 6 1 0 4 5
  2 4 7 1 6 0 3 5
  3 2 6 7 0 1 5 4
  3 5 1 7 0 6 2 4

D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.6, Eq. 32.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
Eugene McDonnell, "Magic Squares and Permutations" APL Quote-Quad 7.3 (Fall, 1976)
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Rosa Maria Pidatella, Hermite and Laguerre Functions: a Unifying Point of View, Università degli Studi di Catania, Sicily, Italy (2019).
Murray Bremner and Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - _N. J. A. Sloane_, Apr 18 2014
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Elliot J. Carr and Matthew J. Simpson, New homogenization approaches for stochastic transport through heterogeneous media, arXiv:1810.08890 [physics.bio-ph], 2018.
W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, Parity reversing involutions on plane trees and 2-Motzkin paths, arXiv:math/0503300 [math.CO], 2005.
Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D14 (1976), 1536-1553.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 8.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 127
S. Goodenough and C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.
S. Goodenough and C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16,
H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 115
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin. 52 (2012), 41-54 (Theorem 1).
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
R. J. Marsh and P. P. Martin, Pascal arrays: counting Catalan sets, arXiv:math/0612572 [math.CO], 2006.
Calin D. Morosan, On the number of broadcast schemes in networks, Information Processing Letters, Volume 100, Issue 5 (2006), 188-193.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Index to divisibility sequences
Index entries for related partition-counting sequences

Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially the same sequence), A000165, A047055, A047657, A084947, A084948, A084949, A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.
Catalan(n-1)*M^(n-1)*n! for M=1,2,3,4,5,6: A001813, A052714 (or A144828), A221954, A052734, A221953, A221955.
Cf. A123023, A059344, A099174, A066325, A001700, A000407, A006882.
Cf. A000085, A135401, A297708. - Manfred Scheucher, Jan 07 2018

List([0..20],n->Factorial(2*n)/Factorial(n)); # Muniru A Asiru, Nov 01 2018

[Factorial(2*n)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 09 2018

A001813 := n->(2*n)!/n!;
A001813 := n -> mul(k, k = select(k-> k mod 4 = 2,[$1 .. 4*n])):
seq(A001813(n), n=0..16);  # Peter Luschny, Jun 23 2011

Table[(2n)!/n!, {n,0,20}] (* Harvey P. Dale, May 02 2011 *)

makelist(binomial(n+n, n)*n!,n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=binomial(n+n,n)*n! \\ Charles R Greathouse IV, Jun 15 2011

first(n) = x='x+O('x^n); Vec(serlaplace((1 - 4*x)^(-1/2))) \\ Iain Fox, Jan 01 2018 (corrected by Iain Fox, Jan 11 2018)

from math import factorial
def A001813(n): return factorial(n<<1)//factorial(n) # Chai Wah Wu, Feb 14 2023

[binomial(2*n,n)*factorial(n) for n in range(0, 17)] # Zerinvary Lajos, Dec 03 2009

E.g.f.: (1-4*x)^(-1/2).
a(n) = (2*n)!/n! = Product_{k=0..n-1} (4*k + 2) = A081125(2*n).
Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)) dx, n >= 0. This representation is unique. - Karol A. Penson, Sep 18 2001
Define a'(1)=1, a'(n) = Sum_{k=1..n-1} a'(n-k)*a'(k)*C(n, k); then a(n)=a'(n+1). - Benoit Cloitre, Apr 27 2003
With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - Paul Barry, May 09 2003
a(n) = A000680(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (4*i + 2) = 4^n*Pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
For asymptotics, see the Robinson paper.
a(k) = (2*k)!/k! = Sum_{i=1..k+1} |A008275(i,k+1)| * k^(i-1). - André F. Labossière, Jun 21 2007
a(n) = 12*A051618(a) n >= 2. - Zerinvary Lajos, Feb 15 2008
a(n) = A000984(n)*A000142(n). - Zerinvary Lajos, Mar 25 2008
a(n) = A016825(n-1)*a(n-1). - Roger L. Bagula, Sep 17 2008
a(n) = (-1)^n*A097388(n). - D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
From Paul Barry, Jan 15 2009: (Start)
G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-... (continued fraction);
a(n) = (n+1)!*A000108(n). (End)
a(n) = Sum_{k=0..n} A132393(n,k)*2^(2n-k). - Philippe Deléham, Feb 10 2009
G.f.: 1/(1-2x-8x^2/(1-10x-48x^2/(1-18x-120x^2/(1-26x-224x^2/(1-34x-360x^2/(1-42x-528x^2/(1-... (continued fraction). - Paul Barry, Nov 25 2009
a(n) = A173333(2*n,n) for n>0; cf. A006963, A001761. - Reinhard Zumkeller, Feb 19 2010
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
  2, 2, 0, 0, 0, 0, ...
  4, 4, 4, 0, 0, 0, ...
  6, 6, 6, 6, 0, 0, ...
  8, 8, 8, 8, 8, 0, ...
  ...
(End)
a(n) = (-2)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0), where Q(k) = 1 + x*(4*k+2) - x*(4*k+4)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 8*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + 1/(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
D-finite with recurrence: a(n) = (4*n-6)*a(n-2) + (4*n-3)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 07 2013
Sum_{n>=0} 1/a(n) = (exp(1/4)*sqrt(Pi)*erf(1/2) + 2)/2 = 1 + A214869, where erf(x) is the error function. - Ilya Gutkovskiy, Nov 10 2016
Sum_{n>=0} (-1)^n/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)), where erfi(x) is the imaginary error function. - Amiram Eldar, Feb 20 2021
a(n) = 1/([x^n] hypergeom([1], [1/2], x/4)). - Peter Luschny, Sep 13 2024
a(n) = 2^n*n!*JacobiP(n, -1/2, -n, 3). - Peter Luschny, Jan 22 2025
G.f.: 2F0(1,1/2;;4x). - R. J. Mathar, Jun 07 2025

More terms from James Sellers, May 01 2000

A047974 a(n) = a(n-1) + 2(n-1)a(n-2).

Original entry on oeis.org

1, 1, 3, 7, 25, 81, 331, 1303, 5937, 26785, 133651, 669351, 3609673, 19674097, 113525595, 664400311, 4070168161, 25330978113, 163716695587, 1075631907655, 7296866339961, 50322142646161, 356790528924523, 2570964805355607, 18983329135883665, 142389639792952801, 1091556096587136051

Offset: 0

N. J. A. Sloane

nonn

Related to partially ordered sets. - Detlef Pauly (dettodet(AT)yahoo.de), Sep 25 2003
The number of partial permutation matrices P in GL_n with P^2=0. Alternatively, the number of orbits of the Borel group of upper triangular matrices acting by conjugation on the set of matrices M in GL_n with M^2=0. - Brian Rothbach (rothbach(AT)math.berkeley.edu), Apr 16 2004
Number of ways to use the elements of {1..n} once each to form a collection of sequences, each having length 1 or 2. - Bob Proctor, Apr 18 2005
Hankel transform is A108400. - Paul Barry, Feb 11 2008
This is also the number of subsets of equivalent ways to arrange the elements of n pairs, when equivalence is defined under the joint operation of (optional) reversal of elements combined with permutation of the labels and the subset maps to itself. - Ross Drewe, Mar 16 2008
Equals inverse binomial transform of A000898. - Gary W. Adamson, Oct 06 2008
a(n) is also the moment of order n for the measure of density exp(-(x-1)^2/4)/(2*sqrt(Pi)) over the interval -oo..oo. - Groux Roland, Mar 26 2011
The n-th term gives the number of fixed-point-free involutions in S_n^B, the group of permutations on the set {-n,...,-1,1,2,...,n}. - Matt Watson, Jul 26 2012
From Peter Bala, Dec 03 2017: (Start)
a(n+k) == a(n) (mod k) for all n and k. Hence for each k, the sequence a(n) taken modulo k is a periodic sequence and the exact period divides k. Cf. A115329.
More generally, the same divisibility property holds for any sequence with an e.g.f. of the form F(x)*exp(x*G(x)), where F(x) and G(x) are power series with integer coefficients and G(0) = 1. See the Bala link for a proof. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Peter Alspaugh, James Garrett, Nataša Jonoska, and Masahico Saito, Structures of Monoids Motivated by DNA Origami, arXiv:2501.14966 [math.RA], 2025. See p. 22.
Tewodros Amdeberhan, Valerio De Angelis, Atul Dixit, Victor H. Moll, and Christophe Vignat, From sequences to polynomials and back, via operator orderings, J. Math. Phys. 54, 123502 (2013); Alternative copy
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k
Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts); Alternative copy
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Discrete Applied Mathematics, Volume 161, Issues 10-11, July 2013, Pages 1378-1394; Alternative copy.
Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011.
Samuele Giraudo, Combalgebraic structures on decorated cliques, arXiv:1709.08416 [math.CO], 2017; and also, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8.
Tom Halverson and Mike Reeks, Gelfand Models for Diagram Algebras, arXiv preprint arXiv:1302.6150 [math.RT], 2013.
Andrei Khruzin, Enumeration of chord diagrams, arXiv:math/0008209 [math.CO], 2000.
G. Latouche and P. G. Taylor, A stochastic fluid model for an ad hoc mobile network, Queueing Syst. 63, No. 1-4, 109-129 (2009), eq. (1).
Robert A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Jocelyn Quaintance and Harris Kwong, Permutations and combinations of colored multisets, JIS 13 (2010) #10.2.6.
Index entries for related partition-counting sequences
Index entries for sequences related to Hermite polynomials

Row sums of A067147.
Column k=2 of A359762.
Cf. A000085, A000680, A000898, A001147, A001813, A100861, A108400, A115329, A132101.
Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), this sequence (q=1), A115329 (q=2), A293720 (q=4).

N = 18; A = zeros(N,1); for n = 1:N; a = factorial(n); s = 0; k = 0; while k <= floor(n/2); b = factorial(n - 2*k); c = factorial(k); s = s + a/(b*c); k = k+1; end; A(n) = s; end; disp(A); % Ross Drewe, Mar 16 2008

[n le 2 select 1 else Self(n-1) + 2*(n-2)*Self(n-2): n in [1..40]]; // G. C. Greubel, Jul 12 2024

seq( add(n!/((n-2*k)!*k!), k=0..floor(n/2)), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 15 2001
with(combstruct):seq(count(([S,{S=Set(Union(Z,Prod(Z,Z)))},labeled],size=n)),n=0..30); # Detlef Pauly (dettodet(AT)yahoo.de), Sep 25 2003
A047974 := n -> I^(-n)*orthopoly[H](n, I/2):
seq(A047974(n), n=0..26); # Peter Luschny, Nov 29 2017

Range[0, 23]!*CoefficientList[ Series[ Exp[x*(1-x^2)/(1 - x)], {x, 0,23 }], x] - (* Zerinvary Lajos, Mar 23 2007 *)
Table[I^(-n)*HermiteH[n, I/2], {n, 0, 23}] - (* Alyssa Byrnes and C. Vignat, Jan 31 2013 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(x^2+x))) \\ Joerg Arndt, May 04 2013

[(-i)^n*hermite(n,i/2) for n in range(41)] # G. C. Greubel, Jul 12 2024

E.g.f.: exp(x^2+x). - Len Smiley, Dec 11 2001
Binomial transform of A001813 (with interpolated zeros). - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} C(k,n-k)*n!/k!. - Paul Barry, Mar 29 2007
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*(2k)!/k!; - Paul Barry, Feb 11 2008
G.f.: 1/(1-x-2*x^2/(1-x-4*x^2/(1-x-6*x^2/(1-x-8*x^2/(1-... (continued fraction). -Paul Barry, Apr 10 2009
E.g.f.: Q(0); Q(k) = 1+(x^2+x)/(2*k+1-(x^2+x)*(2*k+1)/((x^2+x)+(2*k+2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. Cf. A000085 and A115329. - Peter Bala, Dec 07 2011
a(n) ~ 2^(n/2 - 1/2)*exp(sqrt(n/2) - n/2 - 1/8)*n^(n/2). - Vaclav Kotesovec, Oct 08 2012
E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) = 1 + (1+x)/(k+1)/(1-x/(x+1/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
a(n) = i^(-n)*H_{n}(i/2) with i the imaginary unit and H_{n} the Hermite polynomial of degree n. - Alyssa Byrnes and C. Vignat, Jan 31 2013
E.g.f.: -Q(0)/x where Q(k) = 1 - (1+x)/(1 - x/(x - (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013
G.f.: 1/Q(0), where Q(k) = 1 + x*2*k - x/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013
E.g.f.: E(0)-1-x-x^2, where E(k) = 2 + 2*x*(1+x) - 8*k^2 + x^2*(1+x)^2*(2*k+3)*(2*k-1)/E(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 21 2013
E.g.f.: Product_{k>=1} 1/(1 + (-x)^k)^(mu(k)/k). - Ilya Gutkovskiy, May 26 2019
a(n) = Sum_{k=0..floor(n/2)} 2^k*B(n, k), where B are the Bessel numbers A100861. - Peter Luschny, Jun 04 2021

A004211 Shifts one place left under 2nd-order binomial transform.

Original entry on oeis.org

1, 1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, 5120905441, 56878092067, 664920021819, 8155340557697, 104652541401025, 1401572711758403, 19546873773314571, 283314887789276721, 4259997696504874817, 66341623494636864963

Offset: 0

N. J. A. Sloane

Equals the eigensequence of A038207, the square of Pascal's triangle. - Gary W. Adamson, Apr 10 2009
The g.f. of the second binomial transform is 1/(1-2x-x/(1-2x/(1-2x-x/(1-4x/(1-2x-x/(1-6x/(1-2x-x/(1-8x/(1-... (continued fraction). - Paul Barry, Dec 04 2009
Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+2 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=2, otherwise F(k+1)=F(k); see example and Fxtbook link. - Joerg Arndt, Apr 30 2011
It appears that the infinite set of "Shifts 1 place left under N-th order binomial transform" sequences has a production matrix of:
  1, N, 0, 0, 0, ...
  1, 1, N, 0, 0, ...
  1, 2, 1, N, 0, ...
  1, 3, 3, 1, N, ...
  ... (where a diagonal of (N,N,N,...) is appended to the right of Pascal's triangle). a(n) in each sequence is the upper left term of M^n such that N=1 generates A000110, then (N=2 - A004211), (N=3 - A004212), (N=4 - A004213), (N=5 - A005011), ... - Gary W. Adamson, Jul 29 2011
Number of "unlabeled" hierarchical orderings on set partitions of {1..n}, see comments on A034691. - Gus Wiseman, Mar 03 2016
From Lorenzo Sauras Altuzarra, Jun 17 2022: (Start)
Number of n-variate noncontradictory conjunctions of logical equalities of literals (modulo logical equivalence).
Equivalently, number of n-variate noncontradictory Krom formulas with palindromic truth-vector (modulo logical equivalence).
a(n) <= A109457(n). (End)

			From _Joerg Arndt_, Apr 30 2011: (Start)
Restricted growth strings: a(0)=1 corresponds to the empty string;
a(1)=1 to [0]; a(2)=3 to [00], [01], and [02]; a(3) = 11 to
        RGS           F
[ 1]  [ 0 0 0 ]    [ 0 0 0 ]
[ 2]  [ 0 0 1 ]    [ 0 0 0 ]
[ 3]  [ 0 0 2 ]    [ 0 0 2 ]
[ 4]  [ 0 1 0 ]    [ 0 0 0 ]
[ 5]  [ 0 1 1 ]    [ 0 0 0 ]
[ 6]  [ 0 1 2 ]    [ 0 0 2 ]
[ 7]  [ 0 2 0 ]    [ 0 2 2 ]
[ 8]  [ 0 2 1 ]    [ 0 2 2 ]
[ 9]  [ 0 2 2 ]    [ 0 2 2 ]
[10]  [ 0 2 3 ]    [ 0 2 2 ]
[11]  [ 0 2 4 ]    [ 0 2 4 ]. (End)
From _Lorenzo Sauras Altuzarra_, Jun 17 2022: (Start)
The 11 trivariate noncontradictory conjunctions of logical equalities of literals are (x <-> y) /\ (y <-> z), (~ x <-> y) /\ (y <-> z), (x <-> ~ y) /\ (~ y <-> z), (x <-> y) /\ (y <-> ~ z), (x <-> y) /\ (z <-> z), (~ x <-> y) /\ (z <-> z), (x <-> z) /\ (y <-> y), (~ x <-> z) /\ (y <-> y), (y <-> z) /\ (x <-> x), (~ y <-> z) /\ (x <-> x), and (x <-> x) /\ (y <-> y) /\ (z <-> z) (modulo logical equivalence).
The third complete Bell polynomial is x^3 + 3 x y + z; and note that (2^0)^3 + 3*2^0*2^1 + 2^2 = 11.
The truth-vector of (x <-> y) /\ (y <-> z), 10000001, is palindromic. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.5, pp. 366-368
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, On the Limiting Vacillating Tableaux for Integer Sequences, arXiv:2208.13091 [math.CO], 2022.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 22, 29.
David Callan (Proposer), Problem 11567, Amer. Math. Monthly, 118 (2011), 371-378.
Agustina Czenky, Unoriented 2-dimensional TQFTs and the category Rep(S_t wr Z_2), arXiv:2306.08826 [math.QA], 2023. See p. 40.
I. M. Gessel, Applications of the classical umbral calculus, arXiv:math/0108121v3 [math.CO], 2001.
Adam M. Goyt and Lara K. Pudwell, Solution to Monthly Problem 11567, 14 May 2011.
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 20.
Janusz Milek, Quantum Implementation of Risk Analysis-relevant Copulas, arXiv:2002.07389 [stat.ME], 2020.
N. J. A. Sloane, Transforms

Cf. A075497 (row sums).
Cf. A038207.
Cf. A000110 (RGS where s(k) <= F(k) + 1), A004212 (RGS where s(k) <= F(k) + 3), A004213 (s(k) <= F(k) + 4), A005011 (s(k) <= F(k) + 5), A005012 (s(k) <= F(k) + 6), A075506 (s(k) <= F(k) + 7), A075507 (s(k) <= F(k) + 8), A075508 (s(k) <= F(k) + 9), A075509 (s(k) <= F(k) + 10).
Cf. A000587, A009235, A317996, A318179 - A318181.
Main diagonal of A261275.
Cf. A034691, A075729, A109457.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*2^(j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 30 2021
# second Maple program:
a:= n -> CompleteBellB(n, seq(2^k, k=0..n)):
seq(a(n), n=0..23);  # Lorenzo Sauras Altuzarra, Jun 17 2022

Table[ Sum[ StirlingS2[ n, k ] 2^(-k+n), {k, n} ], {n, 16} ] (* Wouter Meeussen *)
Table[2^n BellB[n, 1/2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

a(n):=if n=0 then 1 else sum(2^(n-k)*binomial(n-1,k-1)*a(k-1),k,1,n); /* Vladimir Kruchinin, Nov 28 2011 */

x='x+O('x^66);
egf=exp(intformal(exp(2*x))); /* = 1 + x + 3/2*x^2 + 11/6*x^3 + ... */
/* egf=exp(1/2*(exp(2*x)-1)) */ /* alternative e.g.f. */
Vec(serlaplace(egf))  /* Joerg Arndt, Apr 30 2011 */

def A004211(n): return sum(2^(n-k)*stirling_number2(n, k) for k in (0..n))
print([A004211(n) for n in range(21)]) # Peter Luschny, Apr 15 2020

E.g.f. A(x) satisfies A'(x)/A(x) = e^(2x).
E.g.f.: exp(sinh(x)*exp(x)) = exp(Integral_{t = 0..x} exp(2*t)) = exp((exp(2*x)-1)/2). - Joerg Arndt, Apr 30 2011 and May 13 2011
a(n) = Sum_{k=0..n} 2^(n-k)*Stirling2(n, k). - Emeric Deutsch, Feb 11 2002
G.f.: Sum_{k >= 0} x^k/Product_{j = 1..k} (1-2*j*x). - Ralf Stephan, Apr 18 2004
Stirling transform of A000085. - Vladeta Jovovic May 14 2004
O.g.f.: A(x) = 1/(1-x-2*x^2/(1-3*x-4*x^2/(1-5*x-6*x^2/(1-... -(2*n-1)*x-2*n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n) = e^(-1/2)*2^{n-1}*f_n(1/2). - Milan Janjic, May 30 2008
G.f.: 1/(1-x/(1-2x/(1-x/(1-4x/(1-x/(1-6x/(1-x/(1-8x/(1-... (continued fraction). - Paul Barry, Dec 04 2009
a(n) = upper left term in M^n, M = an infinite square production matrix with an appended diagonal of (2,2,2,...) to the right of Pascal's triangle:
  1, 2, 0, 0, 0, ...
  1, 1, 2, 0, 0, ...
  1, 2, 1, 2, 0, ...
  1, 3, 3, 1, 2, ...
  ... - Gary W. Adamson, Jul 29 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+2*x)*d/dx. Cf. A000110. - Peter Bala, Nov 25 2011
G.f. A(x) satisfies A(x)=1+x/(1-2*x)*A(x/(1-2*x)), a(n) = Sum_{k=1..n} binomial(n-1,k-1)*2^(n-k)*a(k-1), a(0)=1. - Vladimir Kruchinin, Nov 28 2011 [corrected by Ilya Gutkovskiy, May 02 2019]
From Peter Bala, May 16 2012: (Start)
Recurrence equation: a(n+1) = Sum_{k = 0..n} 2^(n-k)*C(n,k)*a(k).
Written umbrally this is a(n+1) = (a + 2)^n (expand the binomial and replace a^k with a(k)). More generally, a*f(a) = f(a+2) holds umbrally for any polynomial f(x). An inductive argument then establishes the umbral recurrence a*(a-2)*(a-4)*...*(a-2*(n-1)) = 1 with a(0) = 1. Compare with the Bell numbers B(n) = A000110(n), which satisfy the umbral recurrence B*(B-1)*...*(B-(n-1)) = 1 with B(0) = 1. Cf. A009235.
Touchard's congruence holds: for odd prime p, a(p+k) == (a(k) + a(k+1)) (mod p) for k = 0,1,2,... (adapt the proof of Theorem 10.1 in Gessel). In particular, a(p) == 2 (mod p) for odd prime p. (End)
G.f.: (2/E(0) - 1)/x  where E(k) = 1 + 1/(1 + 2*x/(1 - 4*(k+1)*x/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
G.f.: (1/E(0)-1)/x  where E(k) = 1 - x/(1 + 2*x - 2*x*(k+1)/E(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Sep 21 2012
a(n) = -1 + 2*Sum_{k=0..n} C(n,k)*A166922(k). - Peter Luschny, Nov 01 2012
G.f.: G(0)- 1/x where G(k) = 1 - (4*x*k-1)/(x - x^4/(x^3 - (4*x*k-1)*(4*x*k+2*x-1)*(4*x*k+4*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 08 2013.
G.f.: (G(0) - 1)/(x-1) where G(k) =  1 - 1/(1-2*k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: -G(0) where G(k) = 1 + 2*(1-k*x)/(2*k*x - 1 - x*(2*k*x - 1)/(x - 2*(1-k*x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 29 2013
G.f.: 1/Q(0), where Q(k) = 1 - x/(1 - 2*x*(2*k+1)/( 1 - x/(1 - 4*x*(k+1)/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Apr 15 2013
G.f.: 1 + x/Q(0), where Q(k)= 1 - x*(2*k+3) - x^2*(2*k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 05 2013
For n > 0, a(n) = exp(-1/2)*Sum_{k > 0} (2*k)^n/(k!*2^k). - Vladimir Reshetnikov, May 09 2013
G.f.: -(1+(2*x+1)/G(0))/x, where G(k)= 2*x*k - x - 1 - 2*(k+1)*x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 20 2013
G.f.: T(0)/(1-x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013
Sum_{k=0..n} C(n,k)*a(k)*a(n-k) = 2^n*Bell(n) = A055882(n). - Vaclav Kotesovec, Apr 03 2016
a(n) ~ 2^n * n^n * exp(n/LambertW(2*n) - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^n). - Vaclav Kotesovec, Jan 07 2019, simplified Oct 01 2022
a(n) = B_n(2^0, ..., 2^(n - 1)), where B_n(x_1, ..., x_n) is the n-th complete Bell polynomial. - Lorenzo Sauras Altuzarra, Jun 17 2022

A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 13, 19, 26, 36, 43, 64, 77, 102, 129, 169, 205, 268, 323, 413, 504, 629, 751, 947, 1131, 1384, 1661, 2024, 2393, 2919, 3442, 4136, 4884, 5834, 6836, 8162, 9531, 11262, 13155, 15493, 17981, 21138, 24472, 28571, 33066, 38475, 44305

Offset: 0

David S. Newman, Nov 21 2004

nonn

From Gus Wiseman, Jan 21 2019: (Start)
Also the number of semistandard Young tableaux where the rows are constant and the entries sum to n. For example, the a(8) = 19 tableaux are:
  8   44   2222   11111111
.
  1   2   11   3   111   22   1111   11   11111   1111   111111
  7   6   6    5   5     4    4      33   3       22     2
.
  1   1   11   111
  2   3   2    2
  5   4   4    3
(End)

			a(5) = 6 because, of the 7 unrestricted partitions of 5, only one, 2 + 2 + 1, has a decreasing sequence of frequencies. Two is used twice, but 1 is used only once.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A100881, A100882, A100884.

Cf. A000085, A000219, A003293, A006951, A100471, A323582.

b:= proc(n, i, t) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i=1, `if`(n>=t, 1, 0), `if`(i=0, 0, b(n, i-1, t)+
       add(b(n-i*j, i-1, j), j=t..floor(n/i))))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 03 2014

b[n_, i_, t_] := b[n, i, t] = If[n<0, 0, If[n == 0, 1, If[i == 1, If[n >= t, 1, 0], If[i == 0, 0, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t, Floor[n/i]}]]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Length/@Split[#]]&]],{n,20}] (* Gus Wiseman, Jan 21 2019 *)

More terms from Vladeta Jovovic, Nov 23 2004

A100861 Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 6, 3, 1, 10, 15, 1, 15, 45, 15, 1, 21, 105, 105, 1, 28, 210, 420, 105, 1, 36, 378, 1260, 945, 1, 45, 630, 3150, 4725, 945, 1, 55, 990, 6930, 17325, 10395, 1, 66, 1485, 13860, 51975, 62370, 10395, 1, 78, 2145, 25740, 135135, 270270, 135135, 1, 91, 3003, 45045, 315315, 945945, 945945, 135135

Offset: 0

Emeric Deutsch, Jan 08 2005

Row n contains 1 + floor(n/2) terms. Row sums yield A000085. T(2n,n) = T(2n-1,n-1) = (2n-1)!! (A001147).
Inverse binomial transform is triangle with T(2n,n) = (2n-1)!!, 0 otherwise. - Paul Barry, May 21 2005
Equivalently, number of involutions of n with k pairs. - Franklin T. Adams-Watters, Jun 09 2006
From Gary W. Adamson, Dec 09 2009: (Start)
If considered as an infinite lower triangular matrix (cf. A144299),
lim_{n->} A100861^n = A118930: (1, 1, 2, 4, 13, 41, ...).
(End)
Sum_{k=0..floor(n/2)} T(n,k)m^(n-2k)s^(2k) is the n-th non-central moment of the normal probability distribution with mean m and standard deviation s. - Stanislav Sykora, Jun 19 2014
Row n is the list of coefficients of the independence polynomial of the n-triangular graph. - Eric W. Weisstein, Nov 11 2016
Restating the 2nd part of the Name, row n is the list of coefficients of the matching-generating polynomial of the complete graph K_n. - Eric W. Weisstein, Apr 03 2018

			T(4, 2) = 3 because in the graph with vertex set {A, B, C, D} and edge set {AB, BC, CD, AD, AC, BD} we have the following three 2-matchings: {AB, CD},{AC, BD} and {AD, BC}.
Triangle starts:
[0] 1;
[1] 1;
[2] 1,  1;
[3] 1,  3;
[4] 1,  6,   3;
[5] 1, 10,  15;
[6] 1, 15,  45,   15;
[7] 1, 21, 105,  105;
[8] 1, 28, 210,  420, 105;
[9] 1, 36, 378, 1260, 945.
.
From _Eric W. Weisstein_, Nov 11 2016: (Start)
As polynomials:
1,
1,
1 + x,
1 + 3*x,
1 + 6*x + 3*x^2,
1 + 10*x + 15*x^2,
1 + 15*x + 45*x^2 + 15*x^3. (End)

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (1983 reprint), 10th edition, 1964, expression 22.3.11 in page 775.
C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.

T. D. Noe, Rows n = 0..100, flattened
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
Gérard Le Caër, A new family of solvable Pearson-Dirichlet random walks, Journal of Statistical Physics 144:1 (2011), pp. 23-45.
Ji Young Choi and Jonathan D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016.
Mikael Fremling, On the modular covariance properties of composite fermions on the torus, arXiv:1810.10391 [cond-mat.str-el], 2018.
Gary R. W. Greaves, Jeven Syatriadi, and Charissa I. Utomo, Chromatic polynomials of signed graphs and dominating-vertex deletion formulae, arXiv:2407.00883 [math.CO], 2024. See p. 11.
A. Hernando, R. Hernando, A. Plastino and A. R. Plastino, The workings of the Maximum Entropy Principle in collective human behavior, arXiv preprint arXiv:1201.0905 [stat.AP], 2012.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.
Aaron Pollack, Exceptional theta functions, arXiv:2211.05280 [math.NT], Nov 2022. See Lemma 7.5.1.
Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Eric Weisstein's World of Mathematics, Triangular Graph
Wikipedia, Normal distribution, section 'Moments'
Jian Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Other versions of this same triangle are given in A144299, A001497, A001498, A111924.
Cf. A000085 (row sums).
Cf. A118930, A008619.

a100861 n k = a100861_tabf !! n !! k
a100861_row n = a100861_tabf !! n
a100861_tabf = zipWith take a008619_list a144299_tabl
-- Reinhard Zumkeller, Jan 02 2014

P[0]:=1: for n from 1 to 14 do P[n]:=sort(expand(P[n-1]+(n-1)*t*P[n-2])) od: for n from 0 to 14 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields the sequence in triangular form
# Alternative:
A100861 := proc(n,k)
    n!/k!/(n-2*k)!/2^k ;
end proc:
seq(seq(A100861(n,k),k=0..n/2),n=0..10) ; # R. J. Mathar, Aug 19 2014

Table[Table[n!/(i! 2^i (n - 2 i)!), {i, 0, Floor[n/2]}], {n, 0, 10}] // Flatten  (* Geoffrey Critzer, Mar 27 2011 *)
CoefficientList[Table[2^(n/2) (-(1/x))^(-n/2) HypergeometricU[-n/2, 1/2, -1/(2 x)], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
CoefficientList[Table[(-I)^n Sqrt[x/2]^n HermiteH[n, I/Sqrt[2 x]], {n, 0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)

T(n,k)=if(k<0 || 2*k>n, 0, n!/k!/(n-2*k)!/2^k) /* Michael Somos, Jun 04 2005 */

T(n, k) = n!/(k!(n-2k)!*2^k).
E.g.f.: exp(z+tz^2/2).
G.f.: g(t, z) satisfies the differential equation g = 1 + zg + tz^2*(d/dz)(zg).
Row generating polynomial = P[n] = [-i*sqrt(t/2)]^n*H(n, i/sqrt(2t)), where H(n, x) is a Hermite polynomial and i=sqrt(-1). Row generating polynomials P[n] satisfy P[0]=1, P[n] = P[n-1] + (n-1)tP[n-2].
T(n, k) = binomial(n, 2k)(2k-1)!!. - Paul Barry, May 21 2002 [Corrected by Roland Hildebrand, Mar 06 2009]
T(n,k) = (n-2k+1)*T(n-1,k-1) + T(n-1,k). - Franklin T. Adams-Watters, Jun 09 2006
E.g.f.: 1 + (x+y*x^2/2)/(E(0)-(x+y*x^2/2)), where E(k) = 1 + (x+y*x^2/2)/(1 + (k+1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 08 2013
T(n,k) = A144299(n,k), k=0..n/2. - Reinhard Zumkeller, Jan 02 2014

A053529 a(n) = n! * number of partitions of n.

Original entry on oeis.org

1, 1, 4, 18, 120, 840, 7920, 75600, 887040, 10886400, 152409600, 2235340800, 36883123200, 628929100800, 11769069312000, 230150688768000, 4833164464128000, 105639166144512000, 2464913876705280000, 59606099200327680000, 1525429559126753280000, 40464026199993876480000

Offset: 0

N. J. A. Sloane, Jan 16 2000

Commuting permutations: number of ordered pairs (g, h) in Sym(n) such that gh = hg.
Equivalently sum of the order of all normalizers of all cyclic subgroups of Sym(n). - Olivier Gérard, Apr 04 2012
From Gus Wiseman, Jan 16 2019: (Start)
Also the number of Young tableaux with distinct entries from 1 to n, where a Young tableau is an array obtained by replacing the dots in the Ferrers diagram of an integer partition of n with positive integers. For example, the a(3) = 18 tableaux are:
  123  213  132  312  231  321
.
  12   21   13   31   23   32
  3    3    2    2    1    1
.
  1  2  1  3  2  3
  2  1  3  1  3  2
  3  3  2  2  1  1
(End)

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.12, solution.

T. D. Noe, Table of n, a(n) for n = 0..200
M. Holloway, M. Shattuck, Commuting pairs of functions on a finite set, PU.M.A, Volume 24 (2013), Issue No. 1.
M. Holloway, M. Shattuck, Commuting pairs of functions on a finite set, Research Gate, 2015.
R. P. Stanley, Pairs with equal squares, Problem 10654, Amer. Math. Monthly, 107 (April 2000), solution p. 368.
Wikipedia, Young tableau

Column k=2 of A362827.
Cf. A000041, A072169, A061256.
Sequences counting pairs of functions from an n-set to itself: A053529, A181162, A239749-A239785, A239836-A239841.
Cf. A000085, A117433, A153452, A296188, A323295, A323434, A323436.

a:= func< n | NumberOfPartitions(n)*Factorial(n) >; [ a(n) : n in [0..25]]; // Vincenzo Librandi, Jan 17 2019

seq(count(Permutation(n))*count(Partition(n)),n=1..20); # Zerinvary Lajos, Oct 16 2006
with(combinat): A053529 := proc(n): n! * numbpart(n) end: seq(A053529(n), n=0..20); # Johannes W. Meijer, Jul 28 2016

Table[PartitionsP[n] n!, {n, 0, 20}] (* T. D. Noe, Jun 19 2012 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, x^k/(1-x^k)/k)))) \\ Joerg Arndt, Apr 16 2010

N=66; x='x+O('x^N); Vec(serlaplace(sum(n=0, N, x^n/prod(k=1,n,1-x^k)))) \\ Joerg Arndt, Jan 29 2011

a(n) = n!*numbpart(n); \\ Michel Marcus, Jul 28 2016

from math import factorial
from sympy import npartitions
def A053529(n): return factorial(n)*npartitions(n) # Chai Wah Wu, Jul 10 2023

E.g.f: Sum_{n>=0} x^n/(Product_{k=1..n} 1-x^k) = exp(Sum_{n>=1} (x^n/n)/(1-x^n)). - Joerg Arndt, Jan 29 2011
a(n) = Sum{k=1..n} (((n-1)!/(n-k)!)*sigma(k)*a(n-k)), n > 0, and a(0)=1. See A274760. - Johannes W. Meijer, Jul 28 2016
a(n) ~ sqrt(Pi/6)*exp(sqrt(2/3)*Pi*sqrt(n))*n^n/(2*exp(n)*sqrt(n)). - Ilya Gutkovskiy, Jul 28 2016

A001470 Number of degree-n permutations of order dividing 3.

Original entry on oeis.org

1, 1, 1, 3, 9, 21, 81, 351, 1233, 5769, 31041, 142011, 776601, 4874013, 27027729, 168369111, 1191911841, 7678566801, 53474964993, 418199988339, 3044269834281, 23364756531621, 199008751634001, 1605461415071823, 13428028220072049, 123280859122040601

Offset: 0

N. J. A. Sloane, J. H. Conway and Simon Plouffe

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Seiichi Manyama, Table of n, a(n) for n = 0..631 (terms 0..100 from T. D. Noe)
Joerg Arndt, Generating Random Permutations, PhD thesis, Australian National University, Canberra, Australia, (2010).
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017. See p. 7.
L. Moser and M. Wyman, On Solutions of x^d = 1 in Symmetric Groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001472.

Column k=3 of A008307.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(x+x^3/3) ))); // G. C. Greubel, Sep 03 2023

spec := [S, {S=Set(Union(Cycle(Z, card=1), Cycle(Z, card=3)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..25) # David Radcliffe, Aug 29 2025

a[n_] := HypergeometricPFQ[{(1-n)/3, (2-n)/3, -n/3}, {}, -9]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 03 2011 *)
With[{nn=30},CoefficientList[Series[Exp[x+x^3/3],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 12 2016 *)

a(n):=n!*sum(if mod(n-k,2)=0 then binomial(k,(3*k-n)/2)*(1/3)^((n-k)/2)/k! else 0,k,floor(n/3),n); /* Vladimir Kruchinin, Sep 07 2010 */

def A001470_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(x+x^3/3) ).egf_to_ogf().list()
A001470_list(40) # G. C. Greubel, Sep 03 2023

a(n) = Sum_{j=0..floor(n/3)} n!/(j!*(n-3j)!*(3^j)) (the latter formula from Roger Cuculière).
E.g.f.: exp(x + (1/3)*x^3).
D-finite with recurrence: a(n) = a(n-1) + (n-1)*(n-2)*a(n-3). - Geoffrey Critzer, Feb 03 2009
a(n) = n!*Sum_{k=floor(n/3)..n, n - k == 0 (mod 2)} binomial(k,(3*k-n)/2)*(1/3)^((n-k)/2)/k!. - Vladimir Kruchinin, Sep 07 2010
a(n) ~ n^(2*n/3)*exp(n^(1/3)-2*n/3)/sqrt(3) * (1 - 1/(6*n^(1/3)) + 25/(72*n^(2/3))). - Vaclav Kotesovec, Jul 28 2013

A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.

Original entry on oeis.org

1, 2, 6, 20, 76, 312, 1384, 6512, 32400, 168992, 921184, 5222208, 30710464, 186753920, 1171979904, 7573069568, 50305536256, 342949298688, 2396286830080, 17138748412928, 125336396368896, 936222729254912, 7136574106003456, 55466948299223040, 439216305474605056, 3540846129311916032

Offset: 0

N. J. A. Sloane

Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).
Also the value of the n-th derivative of exp(x^2) evaluated at 1. - N. Calkin, Apr 22 2010
For n >= 1, a(n) is also the sum of the degrees of the irreducible representations of the group of n X n signed permutation matrices (described in sequence A066051). The similar sum for the "ordinary" symmetric group S_n is in sequence A000085. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 12 2002
It appears that this is also the number of permutations of 1, 2, ..., n+1 such that each term (after the first) is within 2 of some preceding term. Verified for n+1 <= 6. E.g., a(4) = 20 because of the 24 permutations of 1, 2, 3, 4, the only ones not permitted are 1, 4, 2, 3; 1, 4, 3, 2; 4, 1, 2, 3; and 4, 1, 3, 2. - Gerry Myerson, Aug 06 2003
Hankel transform is A108400. - Paul Barry, Feb 11 2008
From Emeric Deutsch, Jun 19 2010: (Start)
Number of symmetric involutions of [2n]. Example: a(2)=6 because we have 1234, 2143, 1324, 3412, 4231, and 4321. See the Egge reference, pp. 419-420.
Number of symmetric involutions of [2n+1]. Example: a(2)=6 because we have 12345, 14325, 21354, 45312, 52341, and 54321. See the Egge reference, pp. 419-420.
(End)
Binomial convolution of sequence A000085: a(n) = Sum_{k=0..n} binomial(n,k)*A000085(k)*A000085(n-k). - Emanuele Munarini, Mar 02 2016
The sequence can be obtained from the infinite product of 2 X 2 matrices [(1,N); (1,1)] by extracting the upper left terms, where N = (1, 3, 5, ...), the odd integers. - Gary W. Adamson, Jul 28 2016
Apparently a(n) is the number of standard domino tableaux of size 2n, where a domino tableau is a generalized Young tableau in which all rows and columns are weakly increasing and all regions are dominos. - Gus Wiseman, Feb 25 2018

			G.f. = 1 + 2*x + 6*x^2 + 20*x^3 + 76*x^4 + 312*x^5 + 1384*x^6 + 6512*x^7 + ...
The a(3) = 20 domino tableaux:
1 1 2 2 3 3
.
1 2 2 3 3
1
.
1 2 3 3   1 1 3 3   1 1 2 2
1 2       2 2       3 3
.
1 1 3 3   1 1 2 2
2         3
2         3
.
1 2 3   1 2 2   1 1 3
1 2 3   1 3 3   2 2 3
.
1 3 3   1 2 2
1       1
2       3
2       3
.
1 2   1 1   1 1
1 2   2 3   2 2
3 3   2 3   3 3
.
1 3   1 2   1 1
1 3   1 2   2 2
2     3     3
2     3     3
.
1 1
2
2
3
3
.
1
1
2
2
3
3 - _Gus Wiseman_, Feb 25 2018

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.1.4 Exer. 31.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 10.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 10.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
R. A. Brualdi, Shi-Mie Ma, Enumeration of involutions by descents and symmetric matrices, Eur. J. Combin. 43 (2015) 220-228
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
C.-O. Chow, Counting involutory, unimodal, and alternating signed permutations, Discr. Math. 306 (2006), 2222-2228.
R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
Eric S. Egge, Restricted symmetric permutations, Ann. Combin., 11 (2007), 405-434.
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv:1302.6150 [math.RT], 2013-2014.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Yen-chi R. Lin, Asymptotic Formula for Symmetric Involutions, arXiv:1310.0988 [math.CO], 2013.
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
D. P. Roberts and A. Venkatesh, Hurwitz monodromy and full number fields, 2014.
Index entries for sequences related to Hermite polynomials

Column k=2 of A294042 and A376826.
Row sums of A122832 and A185296.
Cf. A000085, A000712, A004003, A066051, A099390, A100861, A135401, A138178, A153452, A297388, A299699, A299926, A300056, A300060.

a000898 n = a000898_list !! n
a000898_list = 1 : 2 : (map (* 2) $
   zipWith (+) (tail a000898_list) (zipWith (*) [1..] a000898_list))
-- Reinhard Zumkeller, Oct 10 2011

# For Maple program see A000903.
seq(simplify((-I)^n*HermiteH(n, I)), n=0..25); # Peter Luschny, Oct 23 2015

a[n_] := Sum[ 2^k*StirlingS1[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 17 2011, after Vladeta Jovovic *)
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2(a[n-1]+(n-1)a[n-2])},a,{n,30}] (* Harvey P. Dale, Aug 04 2012 *)
Table[Abs[HermiteH[n, I]], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)
a[ n_] := Sum[ 2^(n - 2 k) n! / (k! (n - 2 k)!), {k, 0, n/2}]; (* Michael Somos, Oct 23 2015 *)

makelist((%i)^n*hermite(n,-%i),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp(2*x + x^2 + x * O(x^n)), n))}; /* Michael Somos, Feb 08 2004 */

{a(n) = if( n<2, max(0, n+1), 2*a(n-1) + (2*n - 2) * a(n-2))}; /* Michael Somos, Feb 08 2004 */

my(x='x+O('x^66)); Vec(serlaplace(exp(2*x+x^2))) \\ Joerg Arndt, Oct 04 2013

{a(n) = sum(k=0, n\2, 2^(n - 2*k) * n! / (k! * (n - 2*k)!))}; /* Michael Somos, Oct 23 2015 */

a(n) = Sum_{m=0..n} |A060821(n,m)| = H(n,-i)*i^n, with the Hermite polynomials H(n,x); i.e., these are row sums of the unsigned triangle A060821.
E.g.f.: exp(x*(x + 2)).
a(n) = 2 * A000902(n) for n >= 1.
a(n) = Sum_{k=0..n} binomial(n,2k)*binomial(2k,k)*k!*2^(n-2k). - N. Calkin, Apr 22 2010
Binomial transform of A047974. - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k). - Vladeta Jovovic, Oct 01 2003
From Paul Barry, Aug 29 2005: (Start)
a(n) = Sum_{k=0..floor(n/2)} A001498(n-k, k) * 2^(n-k).
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2) * 2^((n+k)/2) * (1+(-1)^(n-k))/2. (End)
For asymptotics, see the Robinson paper. [This is disputed by Yen-chi R. Lin. See below, Sep 30 2013.]
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * C(n,2*k) * (2*k)!/k!. - Paul Barry, Feb 11 2008
G.f.: 1/(1 - 2*x - 2*x^2/(1 - 2*x - 4*x^2/(1 - 2*x - 6*x^2/(1 - 2*x - 8*x^2/(1 - ... (continued fraction). - Paul Barry, Feb 25 2010
E.g.f.: exp(x^2 + 2*x) = Q(0); Q(k) = 1 + (x^2 + 2*x)/(2*k + 1 - (x^2 + 2*x)*(2*k + 1)/((x^2 + 2*x) + (2*k + 2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
G.f.: 1/Q(0), where Q(k) = 1 + 2*x*k - x - x/(1 - 2*x*(k + 1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
a(n) = (2*n/e)^(n/2) * exp(sqrt(2*n)) / sqrt(2*e) * (1 + sqrt(2/n)/3 + O(n^(-1))). - Yen-chi R. Lin, Sep 30 2013
0 = a(n)*(2*a(n+1) + 2*a(n+2) - a(n+3)) + a(n+1)*(-2*a(n+1) + a(n+2)) for all n >= 0. - Michael Somos, Oct 23 2015
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k)*B(n, k), where B are the Bessel numbers A100861. - Peter Luschny, Jun 04 2021

More terms from Larry Reeves (larryr(AT)acm.org), Feb 21 2001
Initial condition a(0)=1 added to definition by Jon E. Schoenfield, Oct 01 2013
More terms from Joerg Arndt, Oct 04 2013

cp's OEIS Frontend

A006882 Double factorials n!!: a(n) = n*a(n-2) for n > 1, a(0) = a(1) = 1.

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A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

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A047974 a(n) = a(n-1) + 2*(n-1)*a(n-2).

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A004211 Shifts one place left under 2nd-order binomial transform.

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A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.

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A047974 a(n) = a(n-1) + 2(n-1)a(n-2).

A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.