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 A344501 #6 Apr 21 2024 04:17:03
%S A344501 1,1,2,10,40,176,916,4852,27350,163270,1009396,6504356,43400512,
%T A344501 298682320,2118282440,15433768456,115345136566,882900083222,
%U A344501 6910879999420,55255039432300,450744068706896,3747796352076736,31734090674951512,273414453918459800,2395202886317347900
%N A344501 a(n) = Sum_{k=0..n} binomial(n, k)*HT(n, k) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*HT(n, k), where HT(n, k) is the Hermite triangle A099174.
%F A344501 a(n) = Sum_{j=0..n} even(n - j)*binomial(n, j)*2^((j - n)/2)*n!/(j!*((n - j)/2)!), where even(k) = 1 if k is even and otherwise 0.
%F A344501 From _Vaclav Kotesovec_, Apr 21 2024: (Start)
%F A344501 Recurrence: n*(9*n - 13)*a(n) = (3*n - 4)*(9*n - 5)*a(n-1) + (18*n^3 - 89*n^2 + 131*n - 56)*a(n-2) + (54*n^3 - 219*n^2 + 261*n - 92)*a(n-3) - (n-3)^2*(n-1)*(9*n - 4)*a(n-4).
%F A344501 a(n) ~ n^(n/2 - 3/8) / (2^(3/2) * sqrt(Pi) * exp(n/2 - 2*n^(3/4) + 3*sqrt(n)/4 - 5*n^(1/4)/16 + 1/8)) * (1 + 5351/(5120*n^(1/4))). (End)
%p A344501 a := proc(n) add((if n - j mod 2 = 0 then binomial(n, j)*2^((j - n)/2)*n!/(j!*((n - j)/2)!) else 0 fi), j = 0..n) end: seq(a(n), n = 0..24);
%t A344501 Table[n! * Sum[Binomial[n, 2*j] / (2^j * (n - 2*j)! * j!), {j, 0, n/2}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 21 2024 *)
%Y A344501 Cf. A099174, A344500.
%K A344501 nonn
%O A344501 0,3
%A A344501 _Peter Luschny_, May 22 2021