A346983 Expansion of e.g.f. 1 / (5 - 4 * exp(x))^(1/4).

1, 1, 6, 61, 891, 16996, 400251, 11217781, 364638336, 13486045291, 559192836771, 25691965808026, 1295521405067181, 71131584836353861, 4224255395774155566, 269791923787785076921, 18439806740525320993551, 1342957106015632474616956, 103824389511747541791086511

Offset

0,3

Comments

Stirling transform of A007696.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..360

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A007696(k).

a(n) ~ n! / (Gamma(1/4) * 5^(1/4) * n^(3/4) * log(5/4)^(n + 1/4)). - Vaclav Kotesovec, Aug 14 2021

O.g.f. (conjectural): 1/(1 - x/(1 - 5*x/(1 - 5*x/(1 - 10*x/(1 - 9*x/(1 - 15*x/(1 - ... - (4*n-3)*x/(1 - 5*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type. - Peter Bala, Aug 22 2023

a(0) = 1; a(n) = Sum_{k=1..n} (4 - 3*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

a(0) = 1; a(n) = a(n-1) - 5*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

Maple

g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:
b:= proc(n, m) option remember;
     `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 09 2021

Mathematica

nmax = 18; CoefficientList[Series[1/(5 - 4 Exp[x])^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]

Crossrefs

Cf. A000670, A007696, A305404, A346982, A346984, A346985, A352117, A352118, A352119.

Cf. A094417, A354242, A365567.

Sequence in context: A302535 A086403 A049120 * A271841 A361526 A380039

Adjacent sequences: A346980 A346981 A346982 * A346984 A346985 A346986

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Aug 09 2021

Status

approved