A089464 - OEIS

%I #32 Nov 16 2017 14:12:51

%S 1,4,22,163,1564,18679,268714,4538209,88188280,1940666635,47744244286,

%T 1299383450941,38777402351476,1259552677645903,44247546748659130,

%U 1671904534990870369,67624237153933934704,2915628368081840175379,133499617770334938670198

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

%C 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

%H Alois P. Heinz, <a href="/A089464/b089464.txt">Table of n, a(n) for n = 0..150</a>

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

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

%F a(n) ~ 3*exp(3+exp(-1))*n^(n-1). - _Vaclav Kotesovec_, Jul 08 2013

%p a:= n-> add(3*(n-j+3)^(n-j-1)*binomial(n,j), j=0..n):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 30 2012

%t Table[Sum[3(n-k+3)^(n-k-1) Binomial[n,k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Dec 04 2011 *)

%t CoefficientList[Series[E^x*(-LambertW[-x]/x)^3, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jul 08 2013 *)

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

%Y Cf. A089461, A089463 (triangle).

%Y Column k=3 of A144303.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 05 2003