%I #16 Sep 09 2023 06:51:13

%S 1,2,10,116,2140,52856,1627192,59930480,2568599056,125553289760,

%T 6892279877536,419788155021632,28090704069553600,2048487353985408896,

%U 161687913401407530880,13733087614786273308416,1248892148354210466595072,121073054127693143488709120

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

%C Binomial transform of A008542.

%C 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

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

%F a(n) ~ n! * exp(1/6) * 6^n / (Gamma(1/6) * n^(5/6)). - _Vaclav Kotesovec_, Aug 14 2021

%F a(n+2) = (6*n+8)*a(n+1) - 6*(n+1)*a(n). - _Tani Akinari_, Sep 08 2023

%p g:= proc(n) option remember; `if`(n<2, 1, (6*n-5)*g(n-1)) end:

%p a:= n-> add(binomial(n, k)*g(k), k=0..n):

%p seq(a(n), n=0..17); # _Alois P. Heinz_, Aug 10 2021

%t nmax = 17; CoefficientList[Series[Exp[x]/(1 - 6 x)^(1/6), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Binomial[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 17}]

%t Table[HypergeometricU[1/6, n + 7/6, 1/6]/6^(1/6), {n, 0, 17}]

%o (Maxima) a[n]:=if n<2 then n+1 else (6*n-4)*a[n-1]-6*(n-1)*a[n-2];

%o makelist(a[n],n,0,50); /* _Tani Akinari_, Sep 08 2023 */

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

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Aug 10 2021