A346984 - OEIS

%I #22 Nov 16 2023 11:50:58

%S 1,1,7,85,1495,34477,983983,33476437,1322441575,59492222077,

%T 3002578396255,168005805229285,10321907081030167,690761732852321677,

%U 50015387402165694607,3895721046926471861365,324805103526730206129607,28861947117644330678207389,2722944810091827410698112959

%N Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(1/5).

%C Stirling transform of A008548.

%H Seiichi Manyama, <a href="/A346984/b346984.txt">Table of n, a(n) for n = 0..351</a>

%F a(n) = Sum_{k=0..n} Stirling2(n,k) * A008548(k).

%F a(n) ~ n! / (Gamma(1/5) * 6^(1/5) * n^(4/5) * log(6/5)^(n + 1/5)). - _Vaclav Kotesovec_, Aug 14 2021

%F O.g.f. (conjectural): 1/(1 - x/(1 - 6*x/(1 - 6*x/(1 - 12*x/(1 - 11*x/(1 - 18*x/(1 - ... - (5*n-4)*x/(1 - 6*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type. - _Peter Bala_, Aug 22 2023

%F a(0) = 1; a(n) = Sum_{k=1..n} (5 - 4*k/n) * binomial(n,k) * a(n-k). - _Seiichi Manyama_, Sep 09 2023

%F a(0) = 1; a(n) = a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - _Seiichi Manyama_, Nov 16 2023

%p g:= proc(n) option remember; `if`(n<2, 1, (5*n-4)*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/(6 - 5 Exp[x])^(1/5), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[StirlingS2[n, k] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]

%Y Cf. A000670, A008548, A094418, A305404, A346982, A346983, A346985, A352117, A352118, A352119.

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Aug 09 2021