A292893 Expansion of e.g.f. exp(x * (1 - exp(x))).
1, 0, -2, -3, 8, 55, 84, -637, -4992, -10593, 92060, 1012099, 3642000, -18733585, -354606084, -2157876645, 2003383424, 175455790399, 1766183783868, 5436448194707, -96997103373360, -1770215099996721, -13073420293290148, 22275369715313131
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..556
Programs
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PARI
x='x+O('x^66); Vec(serlaplace(exp(x*(1-exp(x))))) -
PARI
a(n) = n!*sum(k=0, n\2, (-1)^k*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, Jul 09 2022
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PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=2, i, j/(j-1)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022
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PARI
a(n) = sum(k=0, n, (-1)^k*(k+1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022
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PARI
my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(k+1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022
Formula
From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * Stirling2(n-k,k)/(n-k)!.
a(0) = 1; a(n) = -(n-1)! * Sum_{k=2..n} k/(k-1)! * a(n-k)/(n-k)!. (End)
From Seiichi Manyama, Aug 29 2022: (Start)
a(n) = Sum_{k=0..n} (-1)^k * (k+1)^(n-k) * binomial(n,k).
G.f.: Sum_{k>=0} (-x)^k / (1 - (k+1)*x)^(k+1). (End)