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%I #10 Jan 05 2024 07:57:05
%S 1,0,4,2,28,105,686,4718,37864,340611,3406330,37469344,449632492,
%T 5845221941,81833107734,1227496615330,19639945846096,333879079382663,
%U 6009823428889074,114186645148891076,2283732902977823060,47958390962534282489,1055084601175754216782
%N a(n) = n! * (1 + Sum_{k=0..n} (-1)^k * binomial(k+2,3) / k!).
%F a(0) = 1; a(n) = n*a(n-1) + (-1)^n * binomial(n+2,3).
%F a(n) = n! + (-1)^n * A368585(n).
%F E.g.f.: (1 - x * (1-x+x^2/6) * exp(-x)) / (1-x).
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x*sum(k=0, 2, binomial(2, k)*(-x)^k/(k+1)!)*exp(-x))/(1-x)))
%Y Cf. A368765, A368766, A368768.
%Y Cf. A368585, A368763.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jan 04 2024