A368574 a(n) = n! * Sum_{k=0..n} binomial(k+2,3) / k!.
0, 1, 6, 28, 132, 695, 4226, 29666, 237448, 2137197, 21372190, 235094376, 2821132876, 36674727843, 513446190362, 7701692856110, 123227085698576, 2094860456876761, 37707488223782838, 716442276251875252, 14328845525037506580, 300905756025787639951, 6619926632567328080946
Offset: 0
Programs
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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))))
Formula
a(0) = 0; a(n) = n*a(n-1) + binomial(n+2,3).
E.g.f.: x * (1+x+x^2/6) * exp(x) / (1-x).