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A357479 a(n) = (n!/6) * Sum_{k=0..n-3} 1/k!.

%I A357479 #26 Apr 02 2023 14:24:48
%S A357479 0,0,0,1,8,50,320,2275,18256,164388,1644000,18084165,217010200,
%T A357479 2821132886,39495860768,592437911975,9479006592160,161143112067400,
%U A357479 2900576017214016,55110944327067273,1102218886541346600,23146596617368279930,509225125582102160000
%N A357479 a(n) = (n!/6) * Sum_{k=0..n-3} 1/k!.
%H A357479 Seiichi Manyama, <a href="/A357479/b357479.txt">Table of n, a(n) for n = 0..450</a>
%F A357479 a(n) = n! * Sum_{k=0..n} binomial(k,3)/k!.
%F A357479 a(0) = 0; a(n) = n * a(n-1) + binomial(n,3).
%F A357479 E.g.f.: x^3/6 * exp(x)/(1-x).
%F A357479 G.f.: (1/6) * Sum_{k>=3} k! * x^k/(1-x)^(k+1).
%t A357479 Table[n!/6 Sum[1/k!,{k,0,n-3}],{n,0,30}] (* _Harvey P. Dale_, Apr 02 2023 *)
%o A357479 (PARI) a(n) = n!/6*sum(k=0, n-3, 1/k!);
%o A357479 (PARI) a(n) = n!*sum(k=0, n, binomial(k, 3)/k!);
%o A357479 (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(x^3/6*exp(x)/(1-x))))
%o A357479 (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=3, N, k!*x^k/(1-x)^(k+1))/6))
%Y A357479 Column k=3 of A073107.
%Y A357479 Cf. A000522, A007526, A038155, A357480.
%Y A357479 Cf. A000292, A000449.
%K A357479 nonn,easy
%O A357479 0,5
%A A357479 _Seiichi Manyama_, Sep 30 2022