A089461 Hyperbinomial transform of A088957. Also the row sums of triangle A089460, which lists the coefficients for the second hyperbinomial transform.

1, 3, 13, 81, 689, 7553, 101961, 1639529, 30640257, 653150529, 15649353929, 416495026841, 12193949444193, 389572905351425, 13488730646528265, 503205102139969977, 20123584054543823105, 858863606297804378753, 38967500492977755457161, 1872974608860684814735385

Offset: 0

Author

Paul D. Hanna, Nov 05 2003

Keywords

• nonn

Comments

a(n) is also the number of subtrees of the complete graph K_{n+1} which contain a fixed edge. For n=2, the a(2)=3 solutions are the 3 subtrees of complete graph K_3 which contain a fixed edge (i.e. the edge itself and 2 copies of K_{1,2}). - Kellie J. MacPhee, Jul 25 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A088957, A089460 (triangle).

Column k=2 of A144303. - Alois P. Heinz, Oct 30 2012

Programs

Maple

a:= n-> add(2*(n-j+2)^(n-j-1)*binomial(n,j), j=0..n):
seq (a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012

Mathematica

CoefficientList[Series[E^x*(-LambertW[-x]/x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)

PARI

x='x+O('x^50); Vec(serlaplace(exp(x)*(-lambertw(-x)/x)^2)) \\ G. C. Greubel, Nov 16 2017

Formula

a(n) = Sum_{k=0..n} 2*(n-k+2)^(n-k-1)*C(n, k).

E.g.f.: exp(x)*(-LambertW(-x)/x)^2.

a(n) ~ 2*exp(2+exp(-1))*n^(n-1). - Vaclav Kotesovec, Jul 08 2013