A346985 - OEIS

%I #32 Nov 17 2023 11:20:09

%S 1,1,8,113,2325,62896,2109143,84403033,3924963750,207976793991,

%T 12369246804853,815880360117978,59107920881218525,4665585774576259261,

%U 398534278371999103888,36627974592437584634573,3603954453161886215458025,377983931878997401821759456,42095013846928585982896180123

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

%C Stirling transform of A008542.

%C 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

%H Alois P. Heinz, <a href="/A346985/b346985.txt">Table of n, a(n) for n = 0..343</a>

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

%F a(n) ~ n! / (Gamma(1/6) * 7^(1/6) * n^(5/6) * log(7/6)^(n + 1/6)). - _Vaclav Kotesovec_, Aug 14 2021

%F 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

%F 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

%F 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

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

%p b:= proc(n, m) option remember;

%p      `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..18);  # _Alois P. Heinz_, Aug 09 2021

%t nmax = 18; CoefficientList[Series[1/(7 - 6 Exp[x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[StirlingS2[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 18}]

%o (Maxima) a[n]:=if n=0 then 1 else (1/n)*sum(binomial(n,k)*(n+5*k)*a[k],k,0,n-1);

%o makelist(a[n],n,0,50); /* _Tani Akinari_, Aug 22 2023 */

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

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 09 2021