A353404 Expansion of e.g.f. exp(-log(1 - x)^5).
1, 0, 0, 0, 0, 120, 1800, 21000, 235200, 2693880, 34133400, 509823600, 9012207600, 180053908320, 3870208261920, 87083930169600, 2034907999488000, 49491370609706880, 1259740748821328640, 33710658096392887680, 949893399326820528000
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..435
Programs
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Mathematica
With[{nn=20},CoefficientList[Series[Exp[-Log[1-x]^5],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 13 2023 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x)^5))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-log(1-x)^4))) -
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;
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PARI
a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/k!);
Formula
E.g.f.: (1 - x)^(-(log(1 - x))^4).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,5)| * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/k!.