A089464 Hyperbinomial transform of A089461. Also the row sums of triangle A089463, which lists the coefficients for the third hyperbinomial transform.

1, 4, 22, 163, 1564, 18679, 268714, 4538209, 88188280, 1940666635, 47744244286, 1299383450941, 38777402351476, 1259552677645903, 44247546748659130, 1671904534990870369, 67624237153933934704, 2915628368081840175379, 133499617770334938670198

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+2} which contain 2 fixed adjacent edges (i.e. a fixed K_{1,2}). For n=2, the a(2)=4 solutions are the 4 subtrees of K_4 which contain 2 fixed adjacent edges (i.e. those 2 edges, 1 copy of K_{1,3}, and 2 copies of P_4). - Kellie J. MacPhee, Jul 25 2013

Links

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

Crossrefs

Cf. A089461, A089463 (triangle).

Column k=3 of A144303.

Programs

Maple

a:= n-> add(3*(n-j+3)^(n-j-1)*binomial(n,j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012

Mathematica

Table[Sum[3(n-k+3)^(n-k-1) Binomial[n,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Dec 04 2011 *)
CoefficientList[Series[E^x*(-LambertW[-x]/x)^3, {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)^3)) \\ G. C. Greubel, Nov 16 2017

Formula

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

E.g.f.: exp(x)*(-LambertW(-x)/x)^3.

a(n) ~ 3*exp(3+exp(-1))*n^(n-1). - Vaclav Kotesovec, Jul 08 2013