A047974 -id:A047974 - cp's OEIS frontend

A362176 Expansion of e.g.f. exp(x * (1-2*x)).

1, 1, -3, -11, 25, 201, -299, -5123, 3249, 167185, 50221, -6637179, -8846903, 309737689, 769776645, -16575533939, -62762132639, 998072039457, 5265897058909, -66595289781995, -466803466259079, 4860819716300521, 44072310882063157, -383679824152382691

Offset: 0

Seiichi Manyama, Apr 10 2023

Seiichi Manyama, Table of n, a(n) for n = 0..668

Column k=4 of A362277.

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), this sequence (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-2*x^2) ))); // G. C. Greubel, Jul 12 2024

With[{m=30}, CoefficientList[Series[Exp[x-2*x^2], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Jul 12 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-2*x))))

[(-sqrt(2))^n*hermite(n, 1/(2*sqrt(2))) for n in range(31)] # G. C. Greubel, Jul 12 2024

a(n) = a(n-1) - 4*(n-1)*a(n-2) for n > 1.
a(n) = n! * Sum_{k=0..floor(n/2)} (-2)^k / (k! * (n-2*k)!).
a(n) = (-sqrt(2))^n * Hermite(n, 1/(2*sqrt(2))). - G. C. Greubel, Jul 12 2024

A362177 Expansion of e.g.f. exp(x * (1-3*x)).

Original entry on oeis.org

1, 1, -5, -17, 73, 481, -1709, -19025, 52753, 965953, -1882709, -59839889, 64418905, 4372890913, -651783677, -367974620369, -309314089439, 35016249465985, 66566286588763, -3715188655737617, -11303745326856599, 434518893361657441, 1858790804545588915

Offset: 0

Seiichi Manyama, Apr 10 2023

Winston de Greef, Table of n, a(n) for n = 0..628

Column k=6 of A362277.

Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), this sequence (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(x-3*x^2) ))); // G. C. Greubel, Jul 12 2024

With[{m=30}, CoefficientList[Series[Exp[x-3*x^2], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Jul 12 2024 *)

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

[(-sqrt(3))^n*hermite(n, 1/(2*sqrt(3))) for n in range(31)] # G. C. Greubel, Jul 12 2024

a(n) = a(n-1) - 6*(n-1)*a(n-2) for n > 1.
a(n) = n! * Sum_{k=0..floor(n/2)} (-3)^k / (k! * (n-2*k)!).
a(n) = (-sqrt(3))^n * Hermite(n, 1/(2*sqrt(3))). - G. C. Greubel, Jul 12 2024

A108400 a(n) = Product_{k = 0..n} (2^k * k!).

Original entry on oeis.org

1, 2, 16, 768, 294912, 1132462080, 52183852646400, 33664847019245568000, 347485857744891213250560000, 64560982045934655213753964953600000, 239901585047846581083822477336190648320000000

Offset: 0

Philippe Deléham, Jul 02 2005

Hankel transform (see A001906 for definition) of the sequences A000898, A001861, A035009(with first term omitted), A047974, A067147(unsigned version), A083886.
Hankel transform of the sequence with e.g.f. exp(x^2). Also (-1)^C(n+1,2)*a(n) is the Hankel transform of the sequence with e.g.f. exp(-x^2). - Paul Barry, Feb 12 2008
Let T(n,k) = (n+1)^k * (1+(-1)^(n-k))/2, then a(n) = det(T(i,j); 0<=i, j<=n). - Paul Barry, Feb 12 2008

G. C. Greubel, Table of n, a(n) for n = 0..38
M. E. Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A000178, A006125, A074962.

Cf. A000898, A001861, A001906, A035009, A047974, A067147, A083886.

BarnesG:= func< n | (&*[Factorial(j): j in [0..n-2]]) >;
[2^Binomial(n+1,2)*BarnesG(n+2): n in [0..15]]; // G. C. Greubel, Jun 21 2022

Table[Product[k!*2^k, {k,0,n}], {n,0,10}] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[2^Binomial[n+1,2]*BarnesG[n+2], {n,0,15}] (* G. C. Greubel, Jun 21 2022 *)

def barnes_g(n): return product(factorial(j) for j in (0..n-2))
[2^binomial(n+1,2)*barnes_g(n+2) for n in (0..15)] # G. C. Greubel, Jun 21 2022

a(n) = A006125(n+1)*A000178(n).
a(n) = Product_{i=1..n} Product_{j=0..i-1} {2*(i-j)}. - Paul Barry, Aug 02 2008
a(n) ~ 2^((n+1)^2/2) * n^(n^2/2+n+5/12) * Pi^((n+1)/2) / (A * exp(3*n^2/4+n-1/12)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014

A158968 Numerator of Hermite(n, 1/6).

Original entry on oeis.org

1, 1, -17, -53, 865, 4681, -73169, -578717, 8640577, 91975825, -1307797649, -17863446149, 241080488353, 4099584856537, -52313249418065, -1085408633265389, 13039168709612161, 325636855090044193, -3664348770051277073, -109170689819225595605, 1144036589538311163361

Offset: 0

N. J. A. Sloane, Nov 12 2009

G. C. Greubel, Table of n, a(n) for n = 0..450
DLMF, Digital library of mathematical functions, Table 18.9.1 for H_n(x).

Cf. A158811, A158960.

Sequences with e.g.f = exp(x + q*x^2): this sequence (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

[Numerator((&+[(-1)^k*Factorial(n)*(1/3)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 10 2018

Numerator[Table[HermiteH[n,1/6],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011 *)
Table[3^n*HermiteH[n, 1/6], {n,0, 50}] (* G. C. Greubel, Jul 10 2018 *)

a(n)=numerator(polhermite(n,1/6)) \\ Charles R Greathouse IV, Jan 29 2016

[3^n*hermite(n, 1/6) for n in range(31)] # G. C. Greubel, Jul 12 2024

From G. C. Greubel, Jun 02 2018: (Start)
a(n) = 3^n * Hermite(n, 1/6).
E.g.f.: exp(x - 9*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/3)^(n-2*k)/(k!*(n-2*k)!)). (End)
D-finite with recurrence a(n) -a(n-1) +18*(n-1)*a(n-2)=0. - [DLMF] Georg Fischer, Feb 06 2021

A255807 E.g.f.: exp(Sum_{k>=1} k^2 * x^k).

Original entry on oeis.org

1, 1, 9, 79, 841, 10821, 162601, 2777419, 52960209, 1112813641, 25509407401, 632772511911, 16870674740569, 480717000225229, 14568646143888201, 467640968478534691, 15841420612530533281, 564519727866573515409, 21102817266052772063689, 825435163723385398719871

Offset: 0

Vaclav Kotesovec, Mar 07 2015

In general, if e.g.f. = exp(Sum_{k>=1} k^m * x^k) and m>0, then a(n) ~ (m+2)^(-1/2) * Gamma(m+2)^(1/(2*m+4)) * exp((m+2)/(m+1) * Gamma(m+2)^(1/(m+2)) * n^((m+1)/(m+2)) + zeta(-m) - n) * n^(n - 1/(2*m+4)).
It appears that the sequence a(n) taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k: if true, then the sequence a(n) taken modulo k would be periodic with period dividing k. - Peter Bala, Nov 14 2017
The above conjecture is true - see the Bala link. - Peter Bala, Jan 20 2018

G. C. Greubel, Table of n, a(n) for n = 0..250
P. Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A082579, A255819, A000262.

nmax=20; CoefficientList[Series[Exp[Sum[k^2*x^k,{k,1,nmax}]],{x,0,nmax}],x] * Range[0,nmax]!
nn = 20; Range[0, nn]! * CoefficientList[Series[Product[Exp[k^2*x^k], {k, 1, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 21 2016 *)

E.g.f.: exp(x*(1+x)/(1-x)^3).
a(n) ~ 2^(-7/8) * 3^(1/8) * n^(n-1/8) * exp(2^(9/4) * 3^(-3/4) * n^(3/4) - n).
a(n) = n!*y(n) where y(0)=1 and y(n)=(Sum_{k=0..n-1} (n-k)^3*y(k))/n for n>=1. - Benedict W. J. Irwin, Jun 02 2016
a(n) = (4*n-3)*a(n-1) - 2*(n-1)*(3*n-8)*a(n-2) + (n-1)*(n-2)*(4*n-11)*a(n-3) - (n-1)*(n-2)*(n-3)*(n-4)*a(n-4). - Peter Bala, Nov 12 2017
E.g.f.: Product_{k>=1} 1/(1 - x^k)^(J_3(k)/k), where J_3() is the Jordan function (A059376). - Ilya Gutkovskiy, May 25 2019

A359762 Array read by ascending antidiagonals. T(n, k) = n![x^n] exp(x + (k/2) x^2). A generalization of the number of involutions (or of 'telephone numbers').

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 10, 7, 4, 1, 1, 1, 26, 25, 10, 5, 1, 1, 1, 76, 81, 46, 13, 6, 1, 1, 1, 232, 331, 166, 73, 16, 7, 1, 1, 1, 764, 1303, 856, 281, 106, 19, 8, 1, 1, 1, 2620, 5937, 3844, 1741, 426, 145, 22, 9, 1, 1

Offset: 0

Peter Luschny, Jan 14 2023

The array is a generalization of the number of involutions of permutations on n letters, A000085, also known as 'telephone numbers'. According to Bednarz et al. the telephone number interpretation "is due to John Riordan, who noticed that T(n, 1) is the number of connection patterns in a telephone system with n subscribers."

In graph theory, the n-th telephone number is the total number of matchings of a complete graph K_n (see the Wikipedia entry). Assuming a network with k possibilities of connections leads to a network that can be modeled by a complete multigraph K(n, k). The total number of connection patterns in such a network is given by T(n, k).

			Array T(n, k) starts:
  [n\k] 0   1      2        3       4        5        6        7
  --------------------------------------------------------------
  [0] 1,    1,     1,       1,      1,       1,       1,       1, ... [A000012]
  [1] 1,    1,     1,       1,      1,       1,       1,       1, ... [A000012]
  [2] 1,    2,     3,       4,      5,       6,       7,       8, ... [A000027]
  [3] 1,    4,     7,      10,     13,      16,      19,      22, ... [A016777]
  [4] 1,   10,    25,      46,     73,     106,     145,     190, ... [A100536]
  [5] 1,   26,    81,     166,    281,     426,     601,     806, ...
  [6] 1,   76,   331,     856,   1741,    3076,    4951,    7456, ...
  [7] 1,  232,  1303,    3844,   8485,   15856,   26587,   41308, ...
  [8] 1,  764,  5937,   21820,  57233,  123516,  234529,  406652, ...
  [9] 1, 2620, 26785,  114076, 328753,  757756, 1510705, 2719900, ...
   [A000085][A047974][A115327][A115329][A115331]

John Riordan, Introduction to Combinatorial Analysis, Dover (2002).

Urszula Bednarz and Małgorzata Wołowiec-Musiał, On a new generalization of telephone numbers, Turkish Journal of Mathematics: Vol. 43: No. 3, (2019).
Carlos M. da Fonseca and Anthony G. Shannon, Telephone numbers extensions, J. Interdisc. Math. (2025) Vol. 28, No. 4, 1573-1580. See pp. 1574, 1577.
Wikipedia, Telephone numbers.

Rows include: A000012, A000027, A016777, A100536.
Columns include: A000012, A000085, A047974, A115327, A115329, A115331, A293720.
Cf. A000085, A277614, A359760.

T := (n, k) -> add(binomial(n, j)*doublefactorial(j-1)*k^(j/2), j = 0..n, 2):
for n from 0 to 9 do lprint(seq(T(n, k), k = 0..7)) od;
T := (n, k) -> ifelse(k=0, 1, I^(-n)*(2*k)^(n/2)*KummerU(-n/2, 1/2, -1/(2*k))):
seq(seq(simplify(T(n-k, k)), k = 0..n), n = 0..10);
T := proc(n, k) exp(x + (k/2)*x^2): series(%, x, 16): n!*coeff(%, x, n) end:
seq(lprint(seq(simplify(T(n, k)), k = 0..8)), n = 0..9);
T := proc(n, k) option remember; if n = 0 or n = 1 then 1 else T(n, k-1) +
n*(k-1)*T(n, k-2) fi end: for n from 0 to 9 do seq(T(n, k), k=0..9) od;
# Only to check the interpretation as a determinant of a lower Hessenberg matrix:
gen := proc(i, j, n) local ev, tv; ev := irem(j+i, 2) = 0; tv := j < i and not ev;
if j > i + 1 then 0 elif j = i + 1 then -1 elif j <= i and ev then 1
elif tv and i < n then x*(n + 1 - i) - 1 else x fi end:
det := M -> LinearAlgebra:-Determinant(M):
p := (n, k) -> subs(x = k, det(Matrix(n, (i, j) -> gen(i, j, n)))):
for n from 0 to 9 do seq(p(n, k), k = 0..7) od;

T[n_, k_] := Sum[Binomial[n, j] Factorial2[j-1] * If[j==0, 1,  k^(j/2)], {j, 0, n, 2}];
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 25 2023 *)

from math import factorial, comb
def oddfactorial(n: int) -> int:
    return factorial(2 * n) // (2**n * factorial(n))
def T(n: int, k: int) -> int:
    return sum(comb(n, 2 * j) * oddfactorial(j) * k**j for j in range(n + 1))
for n in range(10): print([T(n, k) for k in range(8)])

T(n, k) = Sum_{j=0..n, j even} binomial(n, j) * (j - 1)!! * k^(j/2).
T(n, k) = T(n, k-1) + n*(k-1)*T(n, k-2) for n >= 2, T(n, 0) = T(n, 1) = 1.
T(n, k) = i^(-n) * (2*k)^(n/2) * KummerU(-n/2, 1/2, -1/(2*k)) for k >= 1, and T(n, 0) = 1.

A190877 Expansion of e.g.f. exp(x+x^5).

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 15121, 1844641, 20013841, 119845441, 519072841, 1816454641, 223394731561, 3501661887361, 29675906201761, 177923109591361, 844925253766561, 104750282797418881

Offset: 0

Vladimir Kruchinin, May 23 2011

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..510

Cf. A047974, A118395, A190875.

With[{nn=30},CoefficientList[Series[Exp[x+x^5],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 25 2015 *)

a(n):=n!*sum(binomial(n+(-4)*j,j)/(n+(-4)*j)!,j,0,n/4);

a(n) = if(n<5, 1, a(n-1)+5!*binomial(n-1, 4)*a(n-5)); \\ Seiichi Manyama, Feb 25 2022

a(n) = n! * Sum_{j=0..n/4} binomial(n+(-4)*j,j)/(n+(-4)*j)!.

a(n) = a(n-1) + 5! * binomial(n-1,4) * a(n-5) for n > 4. - Seiichi Manyama, Feb 25 2022

A288268 Expansion of e.g.f.: exp(Sum_{k>=1} (k-1)*x^k/k).

Original entry on oeis.org

1, 0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561, 175721140, 2581284541, 40864292184, 693347907421, 12548540320876, 241253367679185, 4909234733857696, 105394372192969489, 2380337795595885156, 56410454014314490981, 1399496554158060983080

Offset: 0

Seiichi Manyama, Oct 20 2017

Seiichi Manyama, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

Cf. A000296, A002720, A008275, A009940, A052852, A052887, A059114, A288269.

l:= func< n, a, b | Evaluate(LaguerrePolynomial(n, a), b) >;
[1,0]cat[(Factorial(n)/(n-1))*(2*l(n-1,0,-1) - l(n,0,-1)): n in [2..30]]; // G. C. Greubel, Mar 10 2021

a := proc(n) option remember; if n < 3 then [1, 0, 1][n+1] else
-(n^2 - 4*n + 3)*a(n - 2) + (2*n - 2)*a(n - 1) fi end:
seq(a(n), n = 0..21); # Peter Luschny, Feb 20 2022

Table[If[n<2, 1-n, (n!/(n-1))*(2*LaguerreL[n-1, -1] - LaguerreL[n, -1])], {n, 0, 30}] (* G. C. Greubel, Mar 10 2021 *)

{a(n) = n!*polcoeff(exp(sum(k=1, n, (k-1)*x^k/k)+x*O(x^n)), n)}

[1-n if n<2 else (factorial(n)/(n-1))*(2*gen_laguerre(n-1,0,-1) - gen_laguerre(n,0,-1)) for n in (0..30)] # G. C. Greubel, Mar 10 2021

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} (k-1)*a(n-k)/(n-k)! for n > 0.
E.g.f.: (1 - x) * exp(x/(1 - x)). - Ilya Gutkovskiy, Jul 27 2020
a(n) = (n!/(n-1))*( 2*LaguerreL(n-1, -1) - LaguerreL(n, -1) ) with a(0) = 1, a(1) = 0. - G. C. Greubel, Mar 10 2021
a(n) ~ n^(n - 3/4) * exp(-1/2 + 2*sqrt(n) - n) / sqrt(2) * (1 - 65/(48*sqrt(n))). - Vaclav Kotesovec, Mar 10 2021, minor term corrected Dec 01 2021
From Peter Luschny, Feb 20 2022: (Start)
a(n) = n! * Sum_{k=0..n} (-1)^k * LaguerreL(n-k, k-1, -1).
a(n) = 2*(n - 1)*a(n - 1) - (n^2 - 4*n + 3)*a(n - 2) for n >= 3. (End)
From Peter Bala, May 26 2023: (Start)
a(n) = Sum_{k = 0..n} |Stirling1(n,k)|*A000296(k) (follows from the fundamental theorem of Riordan arrays).
Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is purely periodic with the period dividing k. For example, modulo 7 we obtain the purely periodic sequence [1, 0, 1, 4, 0, 3, 2, 1, 0, 1, 4, 0, 3, 2, ...] of period 7. Cf. A047974. (End)
For n>1, a(n) = (2*n*A002720(n-1) - A002720(n))/(n-1). - Vaclav Kotesovec, May 27 2023

A294361 E.g.f.: exp(Sum_{n>=1} sigma(n) * x^n).

Original entry on oeis.org

1, 1, 7, 43, 409, 3841, 50431, 648187, 10347793, 170363809, 3200390551, 62855417131, 1371594161257, 31147757782753, 768384638386639, 19814802390611131, 545309251861956001, 15661899520801953217, 475833949719419469223, 15042718034104688144299

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 14 2017: (Start)
The terms of the sequence appear to be of the form 6*m + 1.
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(7*n+2) == 0 (mod 7); a(11*n+9) == 0 (mod 11); a(13*n+11) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..430
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): A294363 (k=0), this sequence (k=1), A294362 (k=2).

Cf. A000203, A053529, A274804, A061256, A192065.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 04 2018 *)

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

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000203(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(k*x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
a(n) ~ Pi^(1/3) * exp((3*Pi)^(2/3) * n^(2/3)/2 - 3^(1/3) * n^(1/3) / (2*Pi^(2/3)) + 1/24 - 1/(8*Pi^2) - n) * n^(n - 1/6) / 3^(2/3). - Vaclav Kotesovec, Sep 04 2018

A294362 E.g.f.: exp(Sum_{n>=1} sigma_2(n) * x^n).

Original entry on oeis.org

1, 1, 11, 91, 1105, 13841, 230731, 3955771, 80483201, 1738065025, 41800101931, 1070731623611, 29804263624081, 878224530964561, 27672361220570795, 919409968480087771, 32304618825218432641, 1191168445737728717441, 46119903359374012564171

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 14 2017: (Start)
It appears that the sequence taken modulo 10 is periodic with period (1, 1, 1, 1, 5) of length 5.
More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(5*n+4) = 0 (mod 5); a(7*n+3) == 0 (mod 7); a(11*n+2) == 0 (mod 11); a(13*n+3) == 0 (mod 13); a(17*n+4) == 0 (mod 17); a(19*n+12) == 0 (mod 19). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..417
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): A294363 (k=0), A294361 (k=1), this sequence (k=2).

Cf. A001157.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 04 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sigma(k, 2)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A001157(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(k^2*x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
a(n) ~ (3*Zeta(3))^(1/8) * exp(2^(9/4) * Zeta(3)^(1/4) * n^(3/4) / 3^(3/4) - n^(1/4) / (2^(9/4) * 3^(5/4) * Zeta(3)^(1/4)) - n) * n^(n - 1/8) / 2^(7/8). - Vaclav Kotesovec, Sep 04 2018

cp's OEIS Frontend

A362176 Expansion of e.g.f. exp(x * (1-2*x)).

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A362177 Expansion of e.g.f. exp(x * (1-3*x)).

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A108400 a(n) = Product_{k = 0..n} (2^k * k!).

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A158968 Numerator of Hermite(n, 1/6).

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A255807 E.g.f.: exp(Sum_{k>=1} k^2 * x^k).

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A359762 Array read by ascending antidiagonals. T(n, k) = n!*[x^n] exp(x + (k/2) * x^2). A generalization of the number of involutions (or of 'telephone numbers').

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A190877 Expansion of e.g.f. exp(x+x^5).

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A359762 Array read by ascending antidiagonals. T(n, k) = n![x^n] exp(x + (k/2) x^2). A generalization of the number of involutions (or of 'telephone numbers').