A346985 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(1/6).

1, 1, 8, 113, 2325, 62896, 2109143, 84403033, 3924963750, 207976793991, 12369246804853, 815880360117978, 59107920881218525, 4665585774576259261, 398534278371999103888, 36627974592437584634573, 3603954453161886215458025, 377983931878997401821759456, 42095013846928585982896180123

Offset: 0

Author

Ilya Gutkovskiy, Aug 09 2021

Keywords

• nonn

Comments

Stirling transform of A008542.

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..343

Crossrefs

Cf. A000670, A008542, A094419, A305404, A346982, A346983, A346984, A352117, A352118, A352119.

Cf. A094419, A354252, A365555, A365556, A365557.

Programs

Maple

g:= proc(n) option remember; `if`(n<2, 1, (6*n-5)*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/(7 - 6 Exp[x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 18}]

Maxima

a[n]:=if n=0 then 1 else (1/n)*sum(binomial(n,k)*(n+5*k)*a[k],k,0,n-1);
makelist(a[n],n,0,50); /* Tani Akinari, Aug 22 2023 */

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A008542(k).

a(n) ~ n! / (Gamma(1/6) * 7^(1/6) * n^(5/6) * log(7/6)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021

For n > 0, a(n) = (1/n)*Sum_{k=0..n-1} binomial(n,k)*(n+5*k)*a(k). - Tani Akinari, Aug 22 2023

O.g.f. (conjectural): 1/(1 - x/(1 - 7*x/(1 - 7*x/(1 - 14*x/(1 - 13*x/(1 - 21*x/(1 - ... - (6*n-5)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Aug 25 2023

a(0) = 1; a(n) = a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023