This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A377954 #12 Nov 12 2024 09:15:29
%S A377954 1,3,9,31,117,471,2053,9339,45321,227467,1203681,6556023,37316029,
%T A377954 217944351,1321360797,8201728531,52577120913,344433580179,
%U A377954 2321103364921,15960060854607,112534486969221,808555930139623,5942117054417589,44446333314841131
%N A377954 a(n) = n! * Sum_{k=0..n} binomial(k+2,n-k) / k!.
%F A377954 E.g.f.: (1 + x)^2 * exp(x + x^2).
%F A377954 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 A377954 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 A377954 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 A377954 (PARI) a(n) = n!*sum(k=0, n, binomial(k+2, n-k)/k!);
%Y A377954 Cf. A018191, A047974, A377955, A377956.
%K A377954 nonn,easy
%O A377954 0,2
%A A377954 _Seiichi Manyama_, Nov 12 2024