A368585 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(k+2,3) / k!.
0, 1, 2, 4, 4, 15, -34, 322, -2456, 22269, -222470, 2447456, -29369108, 381798859, -5345183466, 80177752670, -1282844041904, 21808348713337, -392550276838926, 7458455259940924, -149169105198816940, 3132551209175157511, -68916126601853463218
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).