%I
%S 1,1,2,10,40,176,916,4852,27350,163270,1009396,6504356,43400512,
%T 298682320,2118282440,15433768456,115345136566,882900083222,
%U 6910879999420,55255039432300,450744068706896,3747796352076736,31734090674951512,273414453918459800,2395202886317347900
%N a(n) = Sum_{k=0..n} binomial(n, k)*HT(n, k) = Sum_{k=0..n} (1)^(nk)*binomial(n, k)*HT(n, k), where HT(n, k) is the Hermite triangle A099174.
%F 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.
%p 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);
%Y Cf. A099174, A344500.
%K nonn
%O 0,3
%A _Peter Luschny_, May 22 2021
