A076729 -id:A076729 - cp's OEIS frontend

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

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

A082590 Expansion of 1/((1 - 2x)sqrt(1 - 4*x)).

Original entry on oeis.org

1, 4, 14, 48, 166, 584, 2092, 7616, 28102, 104824, 394404, 1494240, 5692636, 21785872, 83688344, 322494208, 1246068806, 4825743832, 18726622964, 72798509728, 283443548276, 1105144970992, 4314388905704, 16862208539008, 65972020761116, 258354647959984, 1012627828868072

Offset: 0

Vladeta Jovovic, May 13 2003

nonn

Row sums of A068555 and A112336. - Paul Barry, Sep 04 2005
Hankel transform is 2^n*(-1)^C(n+1,2) (A120617). - Paul Barry, Apr 26 2009
Number of n-lettered words in the alphabet {1, 2, 3, 4} with as many occurrences of the substring (consecutive subword) [1, 2] as of [1, 3]. - N. J. A. Sloane, Apr 08 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Shalosh B. Ekhad and Doron Zeilberger, Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type, arXiv:1112.6207 [math.CO], 2011. See subpages for rigorous derivations of the g.f., the recurrence, asymptotics for this sequence.
Alejandro Erickson and Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.
Y. Kamiyama, On the middle dimensional homology classes of equilateral polygon spaces, arXiv:1507.03161 [math.AT], 2015.

Bisection of A226302.

Cf. A034430, A068555, A076729, A112336.

A082590 := proc(n)
    coeftayl( 1/(1-2*x)/sqrt(1-4*x),x=0,n) ;
end proc: # R. J. Mathar, Nov 06 2013
A082590 := n -> 2^n*JacobiP(n, 1/2, -1 - n, 3):
seq(simplify(A082590(n)), n = 0..26);  # Peter Luschny, Jan 22 2025

CoefficientList[ Series[ 1/((1 - 2*x)*Sqrt[1 - 4*x]), {x, 0, 25}], x] (* Jean-François Alcover, Mar 26 2013 *)
Table[2^(n) JacobiP[n, 1/2, -1-n, 3], {n, 0, 30}] (* Vincenzo Librandi, May 26 2013 *)

a(n) = 2^n*JacobiP(n, 1/2, -1-n, 3).
A034430(n) = (n!/2^n)*a(n). A076729(n) = n!*a(n).
a(n) = Sum_{k=0..n+1} binomial(2*n+2, k) * sin((n - k + 1)*Pi/2). - Paul Barry, Nov 02 2004
From Paul Barry, Sep 04 2005: (Start)
a(n) = Sum_{k=0..n} 2^(n-k)*binomial(2*k, k).
a(n) = Sum_{k=0..n} (2*k)! * (2*(n-k))!/(n!*k!*(n-k)!). (End)
a(n) = Sum_{k=0..n} C(2*n, n)*C(n, k)/C(2*n, 2*k). - Paul Barry, Mar 18 2007
G.f.: 1/(1 - 4*x + 2*x^2/(1 + x^2/(1 - 4*x + x^2/(1 + x^2/(1 - 4*x + x^2/(1 + ... (continued fraction). - Paul Barry, Apr 26 2009
D-finite with recurrence: n*a(n) + 2*(-3*n+1)*a(n-1) + 4*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Dec 03 2012
a(n) ~ 2^(2*n + 1)/sqrt(Pi*n). - Vaclav Kotesovec, Aug 15 2013
a(n) = 2^(n + 1)*Pochhammer(1/2, n+1)*hyper2F1([1/2,-n], [3/2], -1)/n!. - Peter Luschny, Aug 02 2014
a(n) - 2*a(n-1) = A000984(n). - R. J. Mathar, Apr 24 2024
a(n) = 2^n*JacobiP(n, 1/2, -1 - n, 3). - Peter Luschny, Jan 22 2025

A197365 T(n,k) gives the number of permutations of the set [n] that contain k occurrences of the subword (132); irregular array read by rows (n >= 0 and 0 <= k <= max(0, floor((n-1)/2))).

Original entry on oeis.org

1, 1, 2, 5, 1, 16, 8, 63, 54, 3, 296, 368, 56, 1623, 2649, 753, 15, 10176, 20544, 9024, 576, 71793, 172596, 104814, 13572, 105, 562848, 1569408, 1228608, 259968, 7968, 4853949, 15398829, 14824314, 4532034, 306729, 945, 45664896, 162412416, 185991936, 75929856

Offset: 0

Peter Bala, Oct 14 2011

A permutation p(1)p(2)...p(n) in the symmetric group S_n contains the subword (132) if there are 3 consecutive elements p(i)p(j)p(k) that have the same order relations as (132), that is, p(i) < p(j) > p(k) and p(i) < p(k). For the enumeration of permutations containing the subword (123) see A162975.
From Petros Hadjicostas, Nov 05 2019: (Start)
The attached Maple program gives a recurrence for the o.g.f. of each row in terms of u for T(n,k), the number of permutations of [n] containing exactly k occurrences of the consecutive pattern 123...(r+1)(r+3)(r+2) for r >= 0. In the program, t = r + 2. Here, n >= 0 and 0 <= k <= max(0, (n-1)/t).
Using that recurrence we may get any row or column from the irregular triangular array T(n, k) for any r >= 0. (Here r = 0, while in array A264781 we have r = 2.)
The recurrence follows from manipulation of the bivariate o.g.f/e.g.f. 1/W(u,z) = Sum_{n, k >= 0} T(n, k)*u^k*z^n/n!, whose reciprocal W(u,z) is the solution of the o.d.e. in Theorem 3.2 in Elizalde and Noy (2003) (with m = a = r + 1). The number t = r + 2 is the order of the o.d.e. in terms of the variable z.
(End)

			Table begins
.n\k.|......0......1.....2......3
= = = = = = = = = = = = = = = = =
..0..|......1
..1..|......1
..2..|......2
..3..|......5......1
..4..|.....16......8
..5..|.....63.....54.....3
..6..|....296....368....56
..7..|...1623...2649...753....15
..8..|..10176..20544..9024...576
...
T(4,0) = 16: The 16 permutations of S_4 not containing the subword (132) are (1234), (2134), (2314), (3124), (3214), (1342), (2341), (3241), (2413), (3412), (3421), (4123), (4213), (4231), (4312), (4321).
T(4,1) = 8: The 8 permutations of S_4 with 1 occurrence of the subword (132) are 1243, 1324, 1423, 1432, 2143, 2431, 3142, 4132.

Alois P. Heinz, Rows n = 0..100, flattened
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes, 2011.
S. Elizalde and M. Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.
Petros Hadjicostas, General Maple program for a recurrence for the number of permutations of [n] containing exactly k times the consecutive pattern 123...(r+1)(r+3)(r+2) for r >= 0. [Here r = 0 and in the program t = r + 2 = 2.]

Columns k=0-10 give: A111004, A263885, A263886, A263887, A263888, A263889, A263890, A263891, A263892, A263893, A263894.
T(2n+1,n) gives A001147.
T(2n+2,n) gives 2*A076729.
Cf. A162975, A264781 (pattern 12354).

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u-j, o+j-1, 0)*`if`(j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Oct 30 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u-j, o+j-1, 0]*If[jJean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

E.g.f.: 1/(1 - int_{t = 0..z} exp((u-1)*t^2/2!)) = sqrt(1 - u)/(sqrt(1 - u) -sqrt(Pi/2) * erf(z/2*sqrt(1 - u))) = 1 + z + 2*z^2/2! + (5 + u)*z^3/3! + (16 + 8*u)*z^4/4! + ....

n-th row sum = n!. First column is A111004.

A067994 Hermite numbers.

Original entry on oeis.org

1, 0, -2, 0, 12, 0, -120, 0, 1680, 0, -30240, 0, 665280, 0, -17297280, 0, 518918400, 0, -17643225600, 0, 670442572800, 0, -28158588057600, 0, 1295295050649600, 0, -64764752532480000, 0, 3497296636753920000, 0, -202843204931727360000, 0

Offset: 0

Eric W. Weisstein, Feb 07 2002

sign

|a(n)| is the number of sets of ordered pairs of n labeled elements. - Steven Finch, Nov 14 2021
|a(n)| is the number of square roots of any permutation in S_{2n} whose disjoint cycle decomposition consists of n transpositions, n > 0. For n=2, permutation (1,2)(3,4) in S_4 has exactly |a(2)|=2 square roots: (1,3,2,4) and (1,4,2,3). - Luis Manuel Rivera Martínez, Feb 25 2015
Self-convolution gives A076729(n)*(-1)^n interleaved with zeros. - Vladimir Reshetnikov, Oct 11 2016
Named after the French mathematician Charles Hermite (1822-1901). - Amiram Eldar, Jun 06 2021

			From _Steven Finch_, Nov 14 2021: (Start)
|a(4)| = 12 because the sets of ordered pairs for n = 4 are
  {(1,2),(3,4)}, {(2,1),(3,4)}, {(1,2),(4,3)}, {(2,1),(4,3)},
  {(1,3),(2,4)}, {(3,1),(2,4)}, {(1,3),(4,2)}, {(3,1),(4,2)},
  {(1,4),(3,2)}, {(4,1),(3,2)}, {(1,4),(2,3)}, {(4,1),(2,3)}. (End)

G. C. Greubel, Table of n, a(n) for n = 0..730
Giuseppe Dattoli, Subuhi Khan, and Ujair Ahmad, Hermite numbers and new families of polynomials, arXiv:2503.14930 [math.NT], 2025.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Eric Weisstein's World of Mathematics, Hermite Number.
Wikipedia, Hermite number.

Cf. A097388 (same sequence without zeros).
Cf. A001813, A076729, A126869, A081123.
Cf. A101109 (ordered triples instead of ordered pairs).

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 09 2018

A067994 := n -> pochhammer(-n, n/2):
seq(A067994(n), n = 0..31); # Peter Luschny, Nov 14 2021

HermiteH[Range[0,50], 0]
With[{nmax=50}, CoefficientList[Series[Exp[-x^2], {x,0,nmax}],x]*Range[0, nmax]!] (* G. C. Greubel, Jun 09 2018 *)

a(n) = polhermite(n, 0); \\ Michel Marcus, Feb 27 2015

x='x+O('x^30); Vec(serlaplace(exp(-x^2))) \\ G. C. Greubel, Jun 09 2018

E.g.f.: exp(-x^2). - Vladeta Jovovic, Aug 24 2002
a(n) = (-1)^(n/2)*n!/(n/2)! if n is even, 0 otherwise. - Mitch Harris, Feb 01 2006
a(n) = -(2*n-2)*a(n-2). - Alexander Karpov, Jul 24 2017
E.g.f.: U(0) where U(k) = 1 - x^2/((2*k+1) - x^2*(2*k+1)/(x^2 - 2*(k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 23 2012
G.f.: 1/G(0) where G(k) = 1 + 2*x^2*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2012
E.g.f.: E(0)/(1+x) where E(k) = 1 + x/(1 - x/(x - (k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
E.g.f.: E(0)-1, where E(k) = 2 - x^2/(2*k+1 + x^2/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 24 2013
a(2*k) = A097388(k), a(2*k+1) = 0. - Joerg Arndt, Oct 12 2016
From Peter Luschny, Nov 14 2021: (Start)
a(n) = A057077(n)*A126869(n)*A081123(n). In particular, a(n) is divisible by floor(n/2)!.
a(n) = Pochhammer(-n, n/2). (End)

A097388 2n-th derivative of the Gaussian exp(-x^2) evaluated at x=0.

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

Michael Somos, Aug 12 2004

sign

H_n(0) := n-th derivative of exp(-x^2) at x=0. H_(2*n + 1)(0) = 0 if n >= 0. - Michael Somos, Jan 24 2014

Self-convolution gives A076729(n)*(-1)^n. - Vladimir Reshetnikov, Oct 11 2016

			exp(-x^2) = 1 - x^2 + x^4/4 - x^6/6 + ..., (d/dx)^4 exp(-x^2) = 12 - 60x^2 + ... so a(2)=12.
G.f. = 1 - 2*x + 12*x^2 - 120*x^3 + 1680*x^4 - 30240*x^5 + 665280*x^6 + ...

Cf. A001813, A076729.

A097388 := n -> (-2)^n*n!*JacobiP(n, -1/2, -n, 3):
seq(simplify( A097388(n)), n = 0..18);  # Peter Luschny, Jan 22 2025

a[ n_] := If[ n < 0, 0, HermiteH[ 2 n, 0]]; (* Michael Somos, Jan 24 2014 *)

{a(n) = if( n<0, 0, (-1)^n * (2*n)! / n!)};

E.g.f.: Sum_{k>=0} a(k) * x^(2*k) / (2*k)! = exp(-x^2).
a(n) = (-1)^n *(2*n)! / n!.
G.f.: 1/U(0) where U(k) = 1 + x*(2*k+2)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Nov 14 2012
G.f.: 1/Q(0), where Q(k) = 1 - x*(8*k+2) + x*(8*k+4)/(1 - x*(8*k+6) + x*(8*k+8)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
a(n) = (-1)^n * A001813(n).
G.f. = 1 / (1 + 2*x / (1 + 4*x / (1 + 6*x / ... ))). - Michael Somos, Jan 24 2014
a(n) ~ (-1)^n*2^(2*n+1/2)*n^n/exp(n). - Ilya Gutkovskiy, Oct 11 2016
Sum_{n>=0} 1/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)). - Amiram Eldar, Nov 12 2020
From Nikolaos Pantelidis, Jan 08 2021: (Start)
G.f.: 1/G(1), where G(n) = 1+(8*n-6)*x-8*n*(2*n-1)*x^2/G(n+1); (Jacobi continued fraction)
G.f.: 1/(1 + 2*x - 8*x^2/(1 + 10*x - 48*x^2/(1 + 18*x - 120*x^2/(1 + 26*x - 224*x^2/(1 + 34*x - 360*x^2/(1+ 42*x - 448*x^2/(1+ 50*x - 648*x^2/( ...)))))))) (Jacobi continued fraction). (End)
a(n) = (-2)^n*n!*JacobiP(n, -1/2, -n, 3). - Peter Luschny, Jan 22 2025

A127138 Q(1,n), where Q(m,k) is defined in A127080 and A127137.

Original entry on oeis.org

1, 1, -1, -4, 3, 28, -15, -288, 105, 3984, -945, -70080, 10395, 1506240, -135135, -38384640, 2027025, 1133072640, -34459425, -38038533120, 654729075, 1431213235200, -13749310575, -59645279232000, 316234143225, 2726781752217600, -7905853580625, -135661078090137600, 213458046676875

Offset: 0

N. J. A. Sloane, Mar 24 2007

sign

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

G. C. Greubel, Table of n, a(n) for n = 0..500

A001147 interleaved with A076729.
Column 1 of A127080.
Cf. A127137, A127144, A127145.

Q:= proc(n, k) option remember;
      if k<2 then 1
    elif `mod`(k,2)=0 then (n-k+1)*Q(n+1,k-1) - (k-1)*Q(n+2,k-2)
    else ( (n-k+1)*Q(n+1,k-1) - (k-1)*(n+1)*Q(n+2,k-2) )/n
      fi; end;
seq( Q(1, n), n=0..30); # G. C. Greubel, Jan 30 2020

Q[n_, k_]:= Q[n, k]= If[k<2, 1, If[EvenQ[k], (n-k+1)*Q[n+1, k-1] - (k-1)*Q[n + 2, k-2], ((n-k+1)*Q[n+1, k-1] - (k-1)*(n+1)*Q[n+2, k-2])/n]]; Table[Q[1, k], {k,0,30}] (* G. C. Greubel, Jan 30 2020 *)

@CachedFunction
def Q(n,k):
    if (k<2): return 1
    elif (mod(k,2)==0): return (n-k+1)*Q(n+1,k-1) - (k-1)*Q(n+2,k-2)
    else: return ( (n-k+1)*Q(n+1,k-1) - (k-1)*(n+1)*Q(n+2,k-2) )/n
[Q(1,n) for n in (0..30)] # G. C. Greubel, Jan 30 2020

See A127080 for e.g.f.

A233481 Number of singletons (strong fixed points) in pair-partitions.

Original entry on oeis.org

0, 1, 4, 21, 144, 1245, 13140, 164745, 2399040, 39834585, 742940100, 15374360925, 349484058000, 8654336615925, 231842662751700, 6679510641428625, 205916703920928000, 6762863294018456625, 235719416966063530500, 8689887736412502745125

Offset: 0

Wojciech Bozejko, Dec 11 2013

nonn

For h(V) = number of singletons (non-crossing chords) in the pair-partition of 2n-elementary set P_2(2n), let T(2n) = sum_{V in P_2(2n)} h(V).
Elements of the sequence a(n) = T(2n).
a(n) is the number of linear chord diagrams on 2n vertices with one marked chord such that none of the remaining n-1 chords cross the marked chord, see [Young]. - Donovan Young, Aug 11 2020

G. C. Greubel, Table of n, a(n) for n = 0..400
Marek Bozejko and Wojciech Bozejko, Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups, arXiv:1301.2502 [math.PR], 2013.
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Cf. A001147, A034430, A082590, A076729, A336598.

A081054 counts pair-partitions of a fixed size without singletons, i.e., linear chord diagrams with 2n nodes and n arcs in which each arc crosses another arc.

a := n -> 2*n*GAMMA(1/2+n)*hypergeom([1/2,-n+1],[3/2],-1)/sqrt(Pi);
seq(simplify(a(n)), n = 0..19); # Peter Luschny, Dec 16 2013
# Alternative:
u := (z/2)^2: egf := 2*u*exp(u)*hypergeom([1/2], [3/2], u): ser := series(egf, z, 40): seq((2*n)!*coeff(ser, z, 2*n), n = 0..19); # Peter Luschny, Mar 14 2023

Table[Sum[(2 k - 1)!! (2 n - 2 k - 1)!!, {k, 0, n - 1}], {n,0,30}] (* T. D. Noe, Dec 13 2013 *)

def A233481():
    a, b, n = 0, 1, 1
    while True:
        yield a
        n += 1
        a, b = b, n*((3*n-4)*b/(n-1)-(2*n-3)*a)
a = A233481(); [next(a) for i in range(17)]  # Peter Luschny, Dec 14 2013

a(n) = T_{2n} = n*sum_{k=0..(n-1)} (2k-1)!!*(2n-2k-1)!!, where (2n-1)!! = 1*3*5*...*(2n-1).
From Peter Luschny, Dec 16 2013: (Start)
E.g.f.: x/((1-x)*sqrt(1-2*x)).
a(n) = 2*n*Gamma(1/2+n)*2_F_1([1/2,-n+1],[3/2],-1)/sqrt(Pi), where 2_F_1 is the hypergeometric function.
a(n) = n*((3*n-4)*a(n-1)/(n-1)-(2*n-3)*a(n-2)) for n>1.
a(n) = n*A034430(n-1) for n>=1.
a(n+1)/(n+1)! = JacobiP(n, 1/2, -n-1, 3).
2^n*a(n+1)/(n+1)! = A082590(n).
2^n*a(n+1)/(n+1) = A076729(n). (End)
a(n) ~ 2^(n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Dec 20 2013
a(n) = (2*n)! * [z^(2*n)] 2*u*exp(u)*hypergeom([1/2], [3/2], u), where u = (z/2)^2. - Peter Luschny, Mar 14 2023

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