A143399 Expansion of x^k/Product_{t=k..2k} (1-tx) for k=4.

0, 0, 0, 0, 1, 30, 545, 7770, 95781, 1071630, 11192665, 111095490, 1060634861, 9822843030, 88799732385, 787259974410, 6869327386741, 59158464019230, 503954741177705, 4254156112792530, 35637875826743421, 296621138907400230, 2455329298857576625

Offset: 0

Author

Alois P. Heinz, Aug 12 2008

Keywords

• nonn

Comments

a(n) is also the number of forests of 4 labeled rooted trees of height at most 1 with n labels, where any root may contain >= 1 labels.

This gives also the fifth column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See the e.g.f. given below. See also A193685 for Sheffer comments and the hint for the proof in the o.g.f. formula there. - Wolfdieter Lang, Oct 08 2011

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350
• Index entries for sequences related to rooted trees
• Index entries for linear recurrences with constant coefficients, signature (30, -355, 2070, -5944, 6720).

Crossrefs

4th 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(4): seq(a(n), n=0..30);

Mathematica

LinearRecurrence[{30,-355,2070,-5944,6720},{0,0,0,0,1},30] (* Harvey P. Dale, Mar 12 2013 *)

Formula

G.f.: x^4/((1-4x)(1-5x)(1-6x)(1-7x)(1-8x)).

a(n) = 30a(n-1) -355a(n-2) +2070a(n-3) -5944a(n-4) +6720a(n-5).

E.g.f.: exp(4*x)*((exp(x)-1)^4)/4!. - Wolfdieter Lang, Oct 08 2011