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

0,2

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

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

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 *)

Prog

(PARI) x='x+O('x^50); Vec(serlaplace(exp(x)*(-lambertw(-x)/x)^3)) \\ G. C. Greubel, Nov 16 2017

Crossrefs

Cf. A089461, A089463 (triangle).

Column k=3 of A144303.

Sequence in context: A184942 A000779 A053144 * A111343 A302908 A187123

Adjacent sequences: A089461 A089462 A089463 * A089465 A089466 A089467

Keyword

nonn

Author

Paul D. Hanna, Nov 05 2003

Status

approved