A377954 a(n) = n! * Sum_{k=0..n} binomial(k+2,n-k) / k!.
1, 3, 9, 31, 117, 471, 2053, 9339, 45321, 227467, 1203681, 6556023, 37316029, 217944351, 1321360797, 8201728531, 52577120913, 344433580179, 2321103364921, 15960060854607, 112534486969221, 808555930139623, 5942117054417589, 44446333314841131
Offset: 0
Programs
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PARI
a(n) = n!*sum(k=0, n, binomial(k+2, n-k)/k!);
Formula
E.g.f.: (1 + x)^2 * exp(x + x^2).
a(n) = -(n-4)*a(n-1) + 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.
a(n) = ((n^2-7*n+3)*a(n-1) + 2*(n-1)*(n^2-3*n-1)*a(n-2))/(n^2-5*n+3) for n > 1.
a(n) ~ n^(n/2 + 1) * 2^(n/2 - 3/2) / exp(1/8 - sqrt(n/2) + n/2) * (1 + 157/(48*sqrt(2*n))). - Vaclav Kotesovec, Nov 12 2024