A347014 Expansion of e.g.f.: exp(x) / (1 - 6*x)^(1/6).

1, 2, 10, 116, 2140, 52856, 1627192, 59930480, 2568599056, 125553289760, 6892279877536, 419788155021632, 28090704069553600, 2048487353985408896, 161687913401407530880, 13733087614786273308416, 1248892148354210466595072, 121073054127693143488709120

Offset: 0

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

• nonn

Comments

Binomial transform of A008542.

In general, for k >= 1, if e.g.f. = exp(x) / (1 - k*x)^(1/k), then a(n) ~ n! * exp(1/k) * k^n / (Gamma(1/k) * n^(1 - 1/k)). - Vaclav Kotesovec, Aug 14 2021

Crossrefs

Cf. A000522, A008542, A056547, A084262, A346258, A347012, A347013.

Programs

Maple

g:= proc(n) option remember; `if`(n<2, 1, (6*n-5)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..17);  # Alois P. Heinz, Aug 10 2021

Mathematica

nmax = 17; CoefficientList[Series[Exp[x]/(1 - 6 x)^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 17}]
Table[HypergeometricU[1/6, n + 7/6, 1/6]/6^(1/6), {n, 0, 17}]

Maxima

a[n]:=if n<2 then n+1 else (6*n-4)*a[n-1]-6*(n-1)*a[n-2];
makelist(a[n],n,0,50); /* Tani Akinari, Sep 08 2023 */

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A008542(k).

a(n) ~ n! * exp(1/6) * 6^n / (Gamma(1/6) * n^(5/6)). - Vaclav Kotesovec, Aug 14 2021

a(n+2) = (6*n+8)*a(n+1) - 6*(n+1)*a(n). - Tani Akinari, Sep 08 2023