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%I #13 Dec 31 2023 10:22:52
%S 0,1,6,28,132,695,4226,29666,237448,2137197,21372190,235094376,
%T 2821132876,36674727843,513446190362,7701692856110,123227085698576,
%U 2094860456876761,37707488223782838,716442276251875252,14328845525037506580,300905756025787639951,6619926632567328080946
%N a(n) = n! * Sum_{k=0..n} binomial(k+2,3) / k!.
%F a(0) = 0; a(n) = n*a(n-1) + binomial(n+2,3).
%F E.g.f.: x * (1+x+x^2/6) * exp(x) / (1-x).
%o (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 2, binomial(2, k)*x^k/(k+1)!)*exp(x)/(1-x))))
%Y Cf. A007526, A103519, A368575, A368576.
%Y Cf. A000292, A337001.
%K nonn,easy
%O 0,3
%A _Seiichi Manyama_, Dec 31 2023