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%I #7 Oct 11 2020 05:01:08
%S 1,2,8,52,478,5706,83824,1461944,29510268,676549450,17361810016,
%T 492999348348,15345359136232,519525230896322,19005788951346240,
%U 747102849650454256,31404054519248544016,1405608808807797838866
%N Hyperbinomial transform of A089466 and also the inverse hyperbinomial transform of A089468.
%C See A088956 for the definition of the hyperbinomial transform.
%F a(n) = sum(k=0, n, (n-k+1)^(n-k-1)*C(n, k)*A089466(k)). a(n) = sum(k=0, n, -(n-k-1)^(n-k-1)*C(n, k)*A089468(k)). a(n) = sum(m=0, n, sum(j=0, m, C(m, j)*C(n, n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!)).
%F a(n) ~ exp(1/2) * n^n. - _Vaclav Kotesovec_, Oct 11 2020
%t Flatten[{1, Table[Sum[Sum[Binomial[m, j] * Binomial[n, n-m-j] * n^(n-m-j) * (m+j)! / (-2)^j / m!, {j,0,m}], {m,0,n}], {n,1,20}]}] (* _Vaclav Kotesovec_, Oct 11 2020 *)
%o (PARI) a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n,n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!))
%Y Cf. A089466, A089468, A088956.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 08 2003