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Expansion of e.g.f. exp(-x * (1 - x)) / (1 - x)^2.
1

%I #10 Aug 14 2024 23:14:23

%S 1,1,5,17,101,589,4369,35125,323273,3263129,36301661,439023001,

%T 5748342445,80949641317,1220505157481,19615065647549,334764933094289,

%U 6046684538094385,115242737561241013,2311256839666971169,48658040610273601781,1072909782649220737661

%N Expansion of e.g.f. exp(-x * (1 - x)) / (1 - x)^2.

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

%F a(n) = n*a(n-1) + 3*(n-1)*a(n-2) - 2*(n-1)*(n-2)*a(n-3).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*(1-x))/(1-x)^2))

%o (PARI) a(n) = (-1)^n*n!*sum(k=0, n, binomial(k-2, n-k)/k!);

%Y Cf. A375409, A375425.

%Y Cf. A000255, A077880.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 14 2024