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A353404 Expansion of e.g.f. exp(-log(1 - x)^5).

%I A353404 #21 Mar 13 2023 15:15:27
%S A353404 1,0,0,0,0,120,1800,21000,235200,2693880,34133400,509823600,
%T A353404 9012207600,180053908320,3870208261920,87083930169600,
%U A353404 2034907999488000,49491370609706880,1259740748821328640,33710658096392887680,949893399326820528000
%N A353404 Expansion of e.g.f. exp(-log(1 - x)^5).
%H A353404 Seiichi Manyama, <a href="/A353404/b353404.txt">Table of n, a(n) for n = 0..435</a>
%F A353404 E.g.f.: (1 - x)^(-(log(1 - x))^4).
%F A353404 a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,5)| * a(n-k).
%F A353404 a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/k!.
%t A353404 With[{nn=20},CoefficientList[Series[Exp[-Log[1-x]^5],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 13 2023 *)
%o A353404 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x)^5)))
%o A353404 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-log(1-x)^4)))
%o A353404 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i-1, j-1)*abs(stirling(j, 5, 1))*v[i-j+1])); v;
%o A353404 (PARI) a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/k!);
%Y A353404 Column k=5 of A357882.
%Y A353404 Cf. A000142, A353344, A353358.
%Y A353404 Cf. A347004, A353200.
%K A353404 nonn
%O A353404 0,6
%A A353404 _Seiichi Manyama_, May 06 2022