OFFSET
0,6
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..461
FORMULA
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,4) * a(n-k).
a(n) ~ n! / (4*(1 + 2^(-3/4)*3^(-1/4)) * log(1 + 2^(3/4)*3^(1/4))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 07 2022: (Start)
G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/(24^k * Product_{j=1..4*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/24^k. (End)
MATHEMATICA
nmax = 23; CoefficientList[Series[1/(1 - (Exp[x] - 1)^4/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
PROG
(PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^4/4!))) \\ Michel Marcus, Aug 06 2021
(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/(24^k*prod(j=1, 4*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 4, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022
(PARI) a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/24^k); \\ Seiichi Manyama, May 07 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 06 2021
STATUS
approved