A357479 - OEIS

%I #26 Apr 02 2023 14:24:48

%S 0,0,0,1,8,50,320,2275,18256,164388,1644000,18084165,217010200,

%T 2821132886,39495860768,592437911975,9479006592160,161143112067400,

%U 2900576017214016,55110944327067273,1102218886541346600,23146596617368279930,509225125582102160000

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

%H Seiichi Manyama, <a href="/A357479/b357479.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) = n! * Sum_{k=0..n} binomial(k,3)/k!.

%F a(0) = 0; a(n) = n * a(n-1) + binomial(n,3).

%F E.g.f.: x^3/6 * exp(x)/(1-x).

%F G.f.: (1/6) * Sum_{k>=3} k! * x^k/(1-x)^(k+1).

%t Table[n!/6 Sum[1/k!,{k,0,n-3}],{n,0,30}] (* _Harvey P. Dale_, Apr 02 2023 *)

%o (PARI) a(n) = n!/6*sum(k=0, n-3, 1/k!);

%o (PARI) a(n) = n!*sum(k=0, n, binomial(k, 3)/k!);

%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(x^3/6*exp(x)/(1-x))))

%o (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 Column k=3 of A073107.

%Y Cf. A000522, A007526, A038155, A357480.

%Y Cf. A000292, A000449.

%K nonn,easy

%O 0,5

%A _Seiichi Manyama_, Sep 30 2022