A346895 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^4 / 4!).

1, 0, 0, 0, 1, 10, 65, 350, 1771, 10290, 86605, 977350, 11778041, 138208070, 1590920695, 18895490250, 245692484311, 3587464083850, 57397496312585, 966066470023550, 16713560617838581, 297182550111615630, 5500448659383161275, 107267326981597659250

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

Cf. A000670, A330047, A346894, A346920.

Cf. A000453, A327505, A346923, A353775.

Sequence in context: A365525 A327505 A346954 * A291231 A097791 A212259

Adjacent sequences: A346892 A346893 A346894 * A346896 A346897 A346898

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 06 2021

Status

approved