A089467 Hyperbinomial transform of A089466 and also the inverse hyperbinomial transform of A089468.
1, 2, 8, 52, 478, 5706, 83824, 1461944, 29510268, 676549450, 17361810016, 492999348348, 15345359136232, 519525230896322, 19005788951346240, 747102849650454256, 31404054519248544016, 1405608808807797838866, 66741852193123060505728, 3350816586986433907218500, 177352811048578736727396576
Offset: 0
Keywords
Programs
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Mathematica
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 *) -
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!))
Formula
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!.
a(n) ~ exp(1/2) * n^n. - Vaclav Kotesovec, Oct 11 2020
Extensions
More terms from Michel Marcus, Jan 12 2025
Comments