A047974 -id:A047974 - cp's OEIS frontend

A052503 Number of permutations sigma of [2n] without fixed points such that sigma^4 = Id.

1, 1, 9, 105, 2625, 76545, 3440745, 176080905, 12034447425, 922995698625, 87505195602825, 9203114782686825, 1141501848477415425, 155540530213013570625, 24232951756530007115625, 4112826185329479728735625, 781060320618828163499210625

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..280
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 28

Cf. A001472, A261317, A261381.

Bisection of column k=4 of A261430.

m:=40; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x^2/2 + x^4/4) )); [Factorial(2*n-2)*b[2*n-1]: n in [1..Floor((m-2)/2)]]; // G. C. Greubel, May 14 2019

spec := [S,{S=Set(Union(Cycle(Z,card=2),Cycle(Z,card=4)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nmax = 40}, CoefficientList[Series[Exp[x^2*(2 + x^2)/4], {x, 0, nmax}], x]*(Range[0, nmax])!][[1 ;; -1 ;; 2]] (* G. C. Greubel, May 14 2019 *)

x='x+O('x^40); v=Vec(serlaplace( exp(x^2/2 + x^4/4) )); vector(#v\2, n, v[2*n-1]) \\ G. C. Greubel, May 14 2019

m = 40; T = taylor(exp(x^2/2 + x^4/4), x, 0, 2*m+2); [factorial(2*n)*T.coefficient(x, 2*n) for n in (0..m)] # G. C. Greubel, May 14 2019

a(n) = (2n)! * [x^(2n)] exp(x^2/2 + x^4/4).
D-finite with recurrence a(n) +(-2*n+1)*a(n-1) -2*(n-1)*(2*n-1)*(2*n-3)*a(n-2)=0, with a(0)=1, a(1)=1, a(2)=9. - Corrected by R. J. Mathar, Feb 20 2020 to skip zeros.
a(n) = 2^n*Gamma(n+1/2)*A047974(n)/Pi^(1/2). - Mark van Hoeij, Oct 30 2011

A293528 E.g.f.: exp(x * Product_{k>0} (1 + x^k)).

Original entry on oeis.org

1, 1, 3, 13, 97, 741, 7291, 81313, 1027713, 14231017, 220911571, 3730744821, 68096325793, 1339705629133, 28225576881867, 634123159354441, 15127595174135041, 381586517104288593, 10147599723510322723, 283846981316172613597, 8324822922497497733601

Offset: 0

Seiichi Manyama, Oct 11 2017

From Peter Bala, Mar 28 2022: (Start)
The congruence a(n+k) == a(n) (mod k) holds for all n and k.
It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, the sequence taken modulo 10 becomes [1, 1, 3, 3, 7, 1, 1, 3, 3, 7, ...], a purely periodic sequence with period 5.
3 divides a(3*n+2); 13 divides a(13*n+3) and a(13*n+5); 19 divides a(19*n+5), a(19*n+12) and a(19*n+14). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..431
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A000009, A293527, A293840.

nmax = 25; CoefficientList[Series[E^(x*QPochhammer[-1, x]/2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 11 2017 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(x*prod(k=1, N, (1+x^k)))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000009(k-1)*a(n-k)/(n-k)! for n > 0.

A293841 E.g.f.: exp(Sum_{n>=1} nA000009(n)x^n).

Original entry on oeis.org

1, 1, 5, 49, 409, 4841, 66541, 1006825, 17349809, 333948529, 6997459861, 159199648961, 3918175462345, 103227624161689, 2901807752857469, 86684932131301561, 2738566218754961761, 91236821580866560865, 3196113263245038385189

Offset: 0

Seiichi Manyama, Oct 17 2017

nonn

From Peter Bala, Mar 28 2022: (Start)
The congruence a(n+k) == a(n) (mod k) holds for all n and k.
It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, the sequence taken modulo 10 becomes [1, 1, 5, 9, 9, 1, 1, 5, 9, 9, ...], a purely periodic sequence with exact period 5.
5 divides a(5*n+2), 7 divides a(7*n+3); 17 divides a(17*n+7), a(17*n+8) and a(17*n+11). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..418
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A000009, A293528, A293840.

nmax = 20; CoefficientList[Series[E^Sum[k*PartitionsQ[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k^2*A000009(k)*a(n-k)/(n-k)! for n > 0.

A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 19, 25, 1, 1, 1, 9, 37, 97, 81, 1, 1, 1, 11, 61, 241, 581, 331, 1, 1, 1, 13, 91, 481, 1981, 3661, 1303, 1, 1, 1, 15, 127, 841, 4881, 17551, 26335, 5937, 1, 1, 1, 17, 169, 1345, 10001, 55321, 171697, 202049, 26785, 1

Offset: 0

Seiichi Manyama, Mar 06 2023

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  1,   1,    1,    1,     1, ...
  1,  3,   5,    7,    9,    11, ...
  1,  7,  19,   37,   61,    91, ...
  1, 25,  97,  241,  481,   841, ...
  1, 81, 581, 1981, 4881, 10001, ...

Columns k=0..4 give A000012, A047974, A361278, A361279, A361280.
Main diagonal gives A361281.
Cf. A293012.

T(n, k) = n!*sum(j=0, n, binomial(k*j, n-j)/j!);

E.g.f. of column k: exp(x * (1+x)^k).

T(0,k) = 1; T(n,k) = (n-1)! * Sum_{j=1..n} j * binomial(k,j-1) * T(n-j,k)/(n-j)!.

A361568 Expansion of e.g.f. exp(x^3/6 * (1+x)^3).

Original entry on oeis.org

1, 0, 0, 1, 12, 60, 130, 420, 8400, 101080, 781200, 4435200, 37714600, 607807200, 8660652000, 94007313400, 914497584000, 11566931376000, 198256136478400, 3275456501116800, 46558791351072000, 636647461257808000, 10238792220969312000, 194852563745775936000

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A047974, A361567, A361569.

Cf. A115055, A361279.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/6*(1+x)^3)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=3, i, j*binomial(3, j-3)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k)/(6^k * k!).
a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} k * binomial(3,k-3) * a(n-k)/(n-k)!.
a(n) = (n-1)*(n-2)/6 * (3*a(n-3) + 12*(n-3)*a(n-4) + 15*(n-3)*(n-4)*a(n-5) + 6*(n-3)*(n-4)*(n-5)*a(n-6)). -Seiichi Manyama, Jun 16 2024

A083886 Expansion of e.g.f. exp(3x)exp(x^2).

Original entry on oeis.org

1, 3, 11, 45, 201, 963, 4899, 26253, 147345, 862083, 5238459, 32957037, 214117209, 1433320515, 9867008979, 69734001357, 505212273441, 3747124863747, 28418591888235, 220152270759597, 1740363304031721, 14027180742479043, 115176800996769411, 962726355659386125, 8186311912829551281, 70769800810139187843

Offset: 0

Paul Barry, May 09 2003

Binomial transform of A000898.
Hankel transform is A108400. - Paul Barry, Jun 13 2009
a(n) is the number of self-inverse signed permutations of length 2n that are equal to their reverse-complements and avoid the pattern (-2,-1). As a result, a(n) also gives the same thing but for avoiding any one of (-1,-2), (+2,+1) or (+1,+2) instead of (-2,-1) (See the Hardt and Troyka reference). - Justin M. Troyka, Aug 05 2011
a(n) is also the number of skew-symmetric (n,n)-clans, or the number of B-orbits in the symmetric space of type CI, Sp_{2n}(C)/GL_n(C) where B is a Borel subgroup of Sp_{2n}(C). - Aram Bingham, Oct 08 2019

			Since a(2) = 11, there are 11 self-inverse signed permutations of 4 that are equal to their reverse-complements and avoid (-2,-1). Some of these are: (+3,+4,+1,+2), (+4,-2,-3,+1), (-1,+3,+2,-4), (-1,-2,-3,-4). - _Justin M. Troyka_, Aug 05 2011

Michael De Vlieger, Table of n, a(n) for n = 0..717
Aram Bingham and Ozlem Ugurlu, Sects and lattice paths over the Lagrangian Grassmannian, arXiv:1903.07229 [math.CO], 2019.
A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179--217.
A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).

Cf. A001813, A047974.

a = {1, 3}; For[n = 2, n < 26, n++, a = Append[a, 3 a[[n]] + 2 (n - 1) a[[n - 1]]]]; a (* Justin M. Troyka, Aug 05 2011 *)
CoefficientList[Series[Exp[3*x+x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 25 2013 *)
Table[Abs[HermiteH[n, 3 I/2]], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 11 2016 *)

my(x='x+O('x^26)); Vec(serlaplace(exp(3*x)*exp(x^2))) /* Joerg Arndt, Jul 12 2012 */

E.g.f.: exp(3*x+x^2).
From Paul Barry, Jun 13 2009: (Start)
G.f.: 1/(1-3x-2x^2/(1-3x-4x^2/(1-3x-6x^2/(1-3x-8x^2/(1-... (continued fraction);
a(n) = Sum_{k=0..floor(n/2)} C(n,2k) * (2k)! * 3^(n-2k) / k!. (End)
a(n) = i^n*Hermite_H(n, -3i/2), i=sqrt(-1). - Paul Barry, Jun 15 2009
a(0) = 1; a(1) = 3; a(n) = 3*a(n-1) + 2*(n-1)*a(n-2) for n >= 2. - Justin M. Troyka, Aug 05 2011
E.g.f. 1 + (x+3)*x/(G(0)-x^2-3*x) where G(k)= x^2 + 3*x + k + 1 - (x+3)*x*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 12 2012
G.f.: 1/Q(0) where Q(k) = 1 + 2*x*k - 2*x - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
a(n) ~ n^(n/2)*2^(n/2-1/2)*exp(3*sqrt(n/2)-n/2-9/8) * (1+21*sqrt(2)/(32*sqrt(n))). - Vaclav Kotesovec, Jun 25 2013

A193930 E.g.f.: exp(x+x^2+x^3+x^4).

Original entry on oeis.org

1, 1, 3, 13, 73, 381, 2611, 19993, 165873, 1436473, 14004451, 145099461, 1584090553, 18196817653, 223416271443, 2865429498961, 38330181602401, 535448870264433, 7823019065848003, 118402856414023933, 1856454825152993961, 30160691907215561581

Offset: 0

Vladimir Kruchinin, Aug 09 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..524

Cf. A047974, A118589.

I:=[1,1,3,13]; [n le 4 select I[n] else Self(n-1)+2*(n-2)*Self(n-2)+3*(n-2)*(n-3)*Self(n-3)+4*(n-2)*(n-3)*(n-4)*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Oct 12 2014

A193930:=n->`if`(n=0,1,n!*add(add( (-1)^i*binomial(k,k-i)*binomial(n-4*i-1,k-1)/k!, i=0..(n-k)/4), k=1..n)): seq(A193930(n), n=0..20); # Wesley Ivan Hurt, Oct 11 2014
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=1..min(n, 4)))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017

RecurrenceTable[{a[n] == a[n - 1] + 2 (n - 1) a[n - 2] + (n - 1) (n - 2) 3 a[n - 3] + (n - 1) (n - 2) (n - 3) 4 a[n - 4], a[0] == 1,
a[1] == 1, a[2] == 3, a[3] == 13}, a, {n, 0, 20}] (* Geoffrey Critzer, Oct 11 2014 *)
With[{nn=30},CoefficientList[Series[Exp[x+x^2+x^3+x^4],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 20 2021 *)

a(n):=if n=0 then 1 else n!*sum(sum((-1)^i*binomial(k,k-i)*binomial(n-4*i-1,k-1),i,0,(n-k)/4)/k!,k,1,n);

x = y + O(y^50); concat(0, Vec(serlaplace(exp(x+x^2+x^3+x^4)))) \\ Michel Marcus, Oct 12 2014

a(n) = n!*sum(k=1..n, sum(i=0..(n-k)/4, (-1)^i*binomial(k,k-i)*binomial(n-4*i-1,k-1))/k!), n>0, a(0)=1.

a(n) = a(n-1) + 2*(n-1)*a(n-2) + 3*(n-1)(n-2)*a(n-3) + 4*(n-1)*(n-2)*(n-3)*a(n-4). - Geoffrey Critzer, Oct 11 2014

A293527 E.g.f.: exp(x/Product_{k>0} (1 - x^k)).

Original entry on oeis.org

1, 1, 3, 19, 145, 1401, 15331, 198283, 2840769, 45744625, 807769891, 15590922051, 325339538833, 7316871562729, 175934564213955, 4508362093795771, 122558873094082561, 3522465207528093153, 106681726559176156099, 3395601487535927589235, 113287948824653903674641

Offset: 0

Seiichi Manyama, Oct 11 2017

nonn

From Peter Bala, Mar 25 2022: (Start)
The sequence terms are odd. 3 divides a(3*n+2), 5 divides a(5*n+4), 9 divides a(9*n+8), 15 divides a(15*n+14) and 19 divides a(19*n+3).
More generally, the congruence a(n+k) == a(n) (mod k) holds for all n and k.  It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, taken modulo 7 the sequence becomes [1, 1, 3, 5, 5, 1, 1, 1, 3, 3, 5, 5, 1, 1, ...], a purely periodic sequence with period 7. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..423
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A000041, A293528.

nmax = 25; CoefficientList[Series[E^(x/QPochhammer[x, x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 11 2017 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(x/prod(k=1, N, (1-x^k)))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000041(k-1)*a(n-k)/(n-k)! for n > 0.

A349640 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k!.

Original entry on oeis.org

1, 2, 7, 46, 485, 7066, 130987, 2946182, 77923561, 2369742130, 81467904431, 3124302688222, 132237820201357, 6123150708289226, 307903794151741075, 16709463201832993846, 973385368533058021457, 60583668821975488285282, 4012342371757905842648791, 281735471040327667890013070

Offset: 0

Vaclav Kotesovec, Nov 23 2021

nonn

For each positive integer k, the sequence obtained by reducing a(n) modulo k is a periodic sequence with period dividing k. For example, modulo 5 the sequence becomes [1, 2, 2, 1, 0, 1, 2, 2, 1, 0, ...] with period 5. In particular, a(5*n+4) == 0 (mod 5). Cf. A047974. - Peter Bala, Mar 13 2025

Cf. A007317, A064613, A292632, A349603, A349639, A224500.

gf := exp(x)*(1 - sqrt(1 - 4*x))/(2*x): ser := series(gf, x, 24):
seq(n!*coeff(ser, x, n), n = 0..19);
# Alternative:
a := n -> `if`(n < 4, [1, 2, 7, 46][n + 1], ((4*n^2 - 12*n + 8)*a(n - 3) - (8*n^2 - 13*n + 5)*a(n - 2) + 4*n^2*a(n - 1))/(n + 1)):
seq(a(n), n = 0..19);  # Peter Luschny, Nov 23 2021
# Alternative
seq(simplify(hypergeom([-n, 1/2, 1], [2], -4)), n = 0..19); # Peter Bala, Mar 13 2025

Table[Sum[Binomial[n, j]*CatalanNumber[j]*j!, {j, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n,k) * (binomial(2*k,k)/(k+1)) * k!); \\ Michel Marcus, Nov 23 2021

a(n) ~ 2^(2*n + 1/2) * n^(n-1) / exp(n - 1/4).
From Peter Luschny, Nov 23 2021: (Start)
a(n) = n! * [x^n](exp(x)*(1 - sqrt(1 - 4*x))/(2*x)).
a(n) = (4*(n-1)*(n-2)*a(n - 3) - (n-1)*(8*n-5)*a(n - 2) + 4*n^2*a(n - 1))/(n + 1) for n >= 4.
a(n-1) = A224500(n) / n for n >= 1. (End)
a(n) = hypergeom([-n, 1/2, 1], [2], -4). - Peter Bala, Mar 13 2025

A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2z)!/(2^zz!).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 3, 1, 0, 10, 0, 15, 0, 1, 0, 15, 0, 45, 0, 15, 1, 0, 21, 0, 105, 0, 105, 0, 1, 0, 28, 0, 210, 0, 420, 0, 105, 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0, 1, 0, 45, 0, 630, 0, 3150, 0, 4725, 0, 945, 1, 0, 55, 0, 990, 0, 6930, 0, 17325, 0, 10395, 0

Offset: 0

Peter Luschny, Jan 13 2023

The Kummer numbers K(n, k) are a refinement of the oddfactorial numbers (A001147) in the sense that they are the coefficients of polynomials K(n, x) = Sum_{n..k} K(n, k) * x^k that take the value oddfactorial(n) at x = 1. The coefficients of x^n are the aerated oddfactorial numbers A123023.

These numbers appear in many different versions (see the crossrefs). They are the coefficients of the Chebyshev-Hermite polynomials in signed form when ordered in decreasing powers. Our exposition is based on the seminal paper by Kummer, which preceded the work of Chebyshev and Hermite for more than 20 years. They are also referred to as Bessel numbers of the second kind (Mansour et al.) when the odd powers are omitted.

			Triangle K(n, k) starts:
 [0] 1;
 [1] 1, 0;
 [2] 1, 0,  1;
 [3] 1, 0,  3, 0;
 [4] 1, 0,  6, 0,   3;
 [5] 1, 0, 10, 0,  15, 0;
 [6] 1, 0, 15, 0,  45, 0,   15;
 [7] 1, 0, 21, 0, 105, 0,  105, 0;
 [8] 1, 0, 28, 0, 210, 0,  420, 0, 105;
 [9] 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0;

John Riordan, Introduction to Combinatorial Analysis, Dover (2002), pp. 85-86.

Pierre Humbert, Monographie des polynômes de Kummer, Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Serie 5, Volume 1 (1922), pp. 81-92.
E. E. Kummer, Über die hypergeometrische Reihe, Journal für die reine und angewandte Mathematik 15 (1836): 39-83.
T. Mansour, M. Schork and M. Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Ladislav Truksa, Hypergeometric orthogonal systems of polynomials III, Aktuárské vědy, Vol. 2 (1931), No. 4, 177-203, (see p.200).

Variants: Signed version: A073278. Other variants are the irregular triangle A100861 with zeros deleted, A066325 and A099174 with reversed rows, A111924, A144299, A104556.
Cf. A000085, A001147, A123023, A047974, A115329, A002522, A056107, A359739,
A001879, A001464, A005425, A344501, A123022, A359761.

oddfactorial := proc(z) (2*z)! / (2^z*z!) end:
K := (n, k) -> ifelse(irem(k, 2) = 1, 0, binomial(n, k) * oddfactorial(k/2)):
seq(seq(K(n, k), k = 0..n), n = 0..11);
# Alternative, as coefficients of polynomials:
p := (n, x) -> 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)):
seq(print(seq(coeff(simplify(p(n, x)), x, k), k = 0..n)), n = 0 ..9);
# Using the exponential generating function:
egf := exp(x + (t*x)^2 / 2): ser := series(egf, x, 12):
seq(print(seq(coeff(n! * coeff(ser, x, n), t, k), k = 0..n)), n = 0..9);

K[n_, k_] := K[n, k] = Which[OddQ[k], 0, k == 0, 1, n == k, K[n - 1, n - 2], True, K[n - 1, k] n/(n - k)];
Table[K[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 25 2023 *)

from functools import cache
@cache
def K(n: int, k: int) -> int:
    if k %  2: return 0
    if n <  3: return 1
    if n == k: return K(n - 1, n - 2)
    return (K(n - 1, k) * n) // (n - k)
for n in range(10): print([K(n, k) for k in range(n + 1)])

Let p(n, x) = 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)).
p(n, 1) = A000085(n); p(n, sqrt(2)) = A047974(n); p(n, 2) = A115329(n);
p(2, n) = A002522(n) (n >= 1); p(3, n) = A056107(n) (n >= 1);
p(n, n) = A359739(n) (n >= 1); 2^n*p(n, 1/2) = A005425(n).
K(n, k) = [x^k] p(n, x).
K(n, k) = [t^k] (n! * [x^n] exp(x + (t*x)^2 / 2)).
K(n, n) = A123023(n).
K(n, n-1) = A123023(n + 1).
K(2*n, 2*n) = A001147(n).
K(4*n, 2*n) = A359761, the central terms without zeros.
K(2*n+2, 2*n) = A001879.
Sum_{k=0..n} (-1)^n * i^k * K(n, k) = A001464(n), ((the number of even involutions) - (the number of odd involutions) in the symmetric group S_n (Robert Israel)).
Sum_{k=0..n} Sum_{j=0..k} K(n, j) = A000085(n + 1).
For a recursion see the Python program.

cp's OEIS Frontend

A052503 Number of permutations sigma of [2n] without fixed points such that sigma^4 = Id.

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A293528 E.g.f.: exp(x * Product_{k>0} (1 + x^k)).

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A293841 E.g.f.: exp(Sum_{n>=1} n*A000009(n)*x^n).

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A361277 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(k*j,n-j)/j!.

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A361568 Expansion of e.g.f. exp(x^3/6 * (1+x)^3).

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A083886 Expansion of e.g.f. exp(3*x)*exp(x^2).

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A193930 E.g.f.: exp(x+x^2+x^3+x^4).

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A293527 E.g.f.: exp(x/Product_{k>0} (1 - x^k)).

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A293841 E.g.f.: exp(Sum_{n>=1} nA000009(n)x^n).

A083886 Expansion of e.g.f. exp(3x)exp(x^2).

A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2z)!/(2^zz!).