A346924 - OEIS

%I #19 May 09 2022 08:38:25

%S 1,0,0,0,0,1,15,175,1960,22449,269577,3430790,46480830,671260876,

%T 10329270952,169125055736,2940784282800,54182845939104,

%U 1055291277366108,21674715826211532,468366193441002564,10624074081842024496,252432685158931968768,6270222495850552958004

%N Expansion of e.g.f. 1 / (1 + log(1 - x)^5 / 5!).

%H Seiichi Manyama, <a href="/A346924/b346924.txt">Table of n, a(n) for n = 0..444</a>

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,5)| * a(n-k).

%F a(n) ~ n! * 2^(3/5) * 3^(1/5) * exp(2^(3/5)*15^(1/5)*n) / (5^(4/5) * (exp(2^(3/5)*15^(1/5)) - 1)^(n+1)). - _Vaclav Kotesovec_, Aug 08 2021

%F a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/120^k. - _Seiichi Manyama_, May 06 2022

%t nmax = 23; CoefficientList[Series[1/(1 + Log[1 - x]^5/5!), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 5]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-x)^5/5!))) \\ _Michel Marcus_, Aug 07 2021

%o (PARI) a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/120^k); \\ _Seiichi Manyama_, May 06 2022

%Y Cf. A007840, A346921, A346922, A346923.

%Y Cf. A000482, A347004, A353200.

%K nonn

%O 0,7

%A _Ilya Gutkovskiy_, Aug 07 2021