A357480 - OEIS

%I #18 Oct 01 2022 02:06:38

%S 0,0,0,0,1,10,75,560,4550,41076,410970,4521000,54252495,705283150,

%T 9873965101,148109477880,2369751647900,40285778016680,725144004303300,

%U 13777736081766576,275554721635336365,5786649154342069650,127306281395525539615,2928044472097087420000

%N a(n) = (n!/24) * Sum_{k=0..n-4} 1/k!.

%H Seiichi Manyama, <a href="/A357480/b357480.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) = n! * Sum_{k=0..n} binomial(k,4)/k!.

%F a(0) = 0; a(n) = n * a(n-1) + binomial(n,4).

%F E.g.f.: x^4/24 * exp(x)/(1-x).

%F G.f.: (1/24) * Sum_{k>=4} k! * x^k/(1-x)^(k+1).

%o (PARI) a(n) = n!/24*sum(k=0, n-4, 1/k!);

%o (PARI) a(n) = n!*sum(k=0, n, binomial(k, 4)/k!);

%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(x^4/24*exp(x)/(1-x))))

%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=4, N, k!*x^k/(1-x)^(k+1))/24))

%Y Column k=4 of A073107.

%Y Cf. A000522, A007526, A038155, A357479.

%Y Cf. A000332, A000475.

%K nonn,easy

%O 0,6

%A _Seiichi Manyama_, Sep 30 2022