A089462 2nd hyperbinomial transform of A001858.

1, 3, 14, 93, 822, 9193, 125292, 2022555, 37829468, 805712859, 19270873704, 511742870653, 14946235170120, 476314240239633, 16451368229689808, 612254102183085627, 24428043107239133712, 1040281158638494489075

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} 2*(n-k+2)^(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+2)^(n-m-j+1)*(m+j)!/(-2)^j)/m!.

a(n) ~ 2 * exp(5/2) * n^(n-1). - Vaclav Kotesovec, Oct 11 2020

Mathematica

Table[Sum[Sum[Binomial[m, j]*Binomial[n, n - m - j + 1]*(n + 2)^(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+2)^(n-m-j+1)*(m+j)!/(-2)^j)/m!))

Crossrefs

Cf. A001858, A089460.

Sequence in context: A101220 A078456 A195134 * A088342 A358118 A364629

Adjacent sequences: A089459 A089460 A089461 * A089463 A089464 A089465

Keyword

nonn

Author

Paul D. Hanna, Nov 05 2003

Status

approved