A143400 Expansion of x^k/Product_{t=k..2k} (1-tx) for k=5.

0, 0, 0, 0, 0, 1, 45, 1190, 24150, 416451, 6427575, 91549480, 1227283200, 15695180501, 193333245105, 2310273772170, 26927270656650, 307413790470151, 3449088814306635, 38132767214613260, 416342920938136500, 4497187699884973401, 48129773048982636165

Offset: 0

Author

Alois P. Heinz, Aug 12 2008

Keywords

• nonn

Comments

a(n) is also the number of forests of 5 labeled rooted trees of height at most 1 with n labels, where any root may contain >= 1 labels.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Index entries for sequences related to rooted trees
• Index entries for linear recurrences with constant coefficients, signature (45, -835, 8175, -44524, 127860, -151200).

Crossrefs

5th column of A143395.

Programs

Maple

a := proc(k::nonnegint) local M; M := Matrix(k+1, (i,j)-> if (i=j-1) then 1 elif j=1 then [seq(-1* coeff(product(1-t*x, t=k..2*k), x,u), u=1..k+1)][i] else 0 fi); p-> (M^p)[1,k+1] end(5); seq(a(n), n=0..30);

Mathematica

CoefficientList[Series[x^5/((1-5x)(1-6x)(1-7x)(1-8x)(1-9x)(1-10x)),{x,0,30}],x] (* or *) LinearRecurrence[{45,-835,8175,-44524,127860,-151200},{0,0,0,0,0,1},30] (* Harvey P. Dale, Aug 30 2018 *)

Formula

G.f.: x^5/((1-5x)(1-6x)(1-7x)(1-8x)(1-9x)(1-10x)).

E.g.f.: exp(5*x)*((exp(x)-1)^5)/5!.