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

0,2

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

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

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 *)

Prog

(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!))

Crossrefs

Cf. A001858, A089462, A089463.

Sequence in context: A067545 A004041 A220353 * A220214 A106174 A056814

Adjacent sequences: A089462 A089463 A089464 * A089466 A089467 A089468

Keyword

nonn

Author

Paul D. Hanna, Nov 05 2003

Status

approved