A089465 3rd hyperbinomial transform of A001858; also the hyperbinomial transform of A089462.

1, 4, 23, 178, 1763, 21504, 313585, 5342068, 104376201, 2304582544, 56807530871, 1547599725720, 46202052688603, 1500629138909632, 52697989385197137, 1990117967149595824, 80440669725095395025, 3465573101368534916928

Offset: 0

Author

Paul D. Hanna, Nov 05 2003

Keywords

• nonn

Comments

A001858 enumerates forests of labeled trees with n nodes and shifts 1 place left under the hyperbinomial transform.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..385

Crossrefs

Cf. A001858, A089462, A089463.

Programs

Mathematica

Table[Sum[Sum[Binomial[m, j]*Binomial[n, n - m - j + 1]*(n + 3)^(n - m - j + 1)*(m + j)!/(-2)^j, {j, 0, m}]/m!, {m, 0, n + 1}], {n, 0, 50}] (* G. C. Greubel, Nov 18 2017 *)

PARI

a(n)=if(n<0,0,sum(m=0,n+1,sum(j=0,m,binomial(m,j)*binomial(n,n-m-j+1)*(n+3)^(n-m-j+1)*(m+j)!/(-2)^j)/m!))

Formula

a(n) = Sum_{k=0..n} 3*(n-k+3)^(n-k-1)*C(n, k)*A001858(k).

a(n) = Sum_{m=0..(n+1)} ( Sum_{j=0..m} C(m, j)*C(n, n-m-j+1)*(n+3)^(n-m-j+1)*(m+j)!/(-2)^j )/m!.

a(n) ~ 3 * exp(7/2) * n^(n-1). - Vaclav Kotesovec, Oct 11 2020