A377954 - OEIS

%I #12 Nov 12 2024 09:15:29

%S 1,3,9,31,117,471,2053,9339,45321,227467,1203681,6556023,37316029,

%T 217944351,1321360797,8201728531,52577120913,344433580179,

%U 2321103364921,15960060854607,112534486969221,808555930139623,5942117054417589,44446333314841131

%N a(n) = n! * Sum_{k=0..n} binomial(k+2,n-k) / k!.

%F E.g.f.: (1 + x)^2 * exp(x + x^2).

%F a(n) = -(n-4)*a(n-1) + 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2.

%F 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.

%F 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

%o (PARI) a(n) = n!*sum(k=0, n, binomial(k+2, n-k)/k!);

%Y Cf. A018191, A047974, A377955, A377956.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Nov 12 2024