A292893 - OEIS

%I #41 Feb 23 2023 18:40:14

%S 1,0,-2,-3,8,55,84,-637,-4992,-10593,92060,1012099,3642000,-18733585,

%T -354606084,-2157876645,2003383424,175455790399,1766183783868,

%U 5436448194707,-96997103373360,-1770215099996721,-13073420293290148,22275369715313131

%N Expansion of e.g.f. exp(x * (1 - exp(x))).

%H Seiichi Manyama, <a href="/A292893/b292893.txt">Table of n, a(n) for n = 0..556</a>

%F From _Seiichi Manyama_, Jul 09 2022: (Start)

%F a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * Stirling2(n-k,k)/(n-k)!.

%F a(0) = 1; a(n) = -(n-1)! * Sum_{k=2..n} k/(k-1)! * a(n-k)/(n-k)!. (End)

%F From _Seiichi Manyama_, Aug 29 2022: (Start)

%F a(n) = Sum_{k=0..n} (-1)^k * (k+1)^(n-k) * binomial(n,k).

%F G.f.: Sum_{k>=0} (-x)^k / (1 - (k+1)*x)^(k+1). (End)

%o (PARI) x='x+O('x^66); Vec(serlaplace(exp(x*(1-exp(x)))))

%o (PARI) a(n) = n!*sum(k=0, n\2, (-1)^k*stirling(n-k, k, 2)/(n-k)!); \\ _Seiichi Manyama_, Jul 09 2022

%o (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

%o (PARI) a(n) = sum(k=0, n, (-1)^k*(k+1)^(n-k)*binomial(n, k)); \\ _Seiichi Manyama_, Aug 29 2022

%o (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

%Y Column k=1 of A292894.

%Y Cf. A000587, A052506, A080108.

%K sign

%O 0,3

%A _Seiichi Manyama_, Sep 26 2017