A143404 Expansion of x^k/Product_{t=k..2k} (1-tx) for k=9.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 135, 10065, 547965, 24336312, 934863930, 32189799070, 1017281878470, 30001945084683, 835898091070185, 22206607023852615, 566594907018764715, 13964270139973201114, 333991935681805199700, 7781827783346875932300

Offset: 0

Author

Alois P. Heinz, Aug 12 2008

Keywords

• nonn

Comments

a(n) is also the number of forests of 9 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 (135, -8160, 290790, -6765213, 107358615, -1176812090, 8797620060, -42924478536, 123418922400, -158789030400).

Crossrefs

9th 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(9): seq(a(n), n=0..30);

Mathematica

CoefficientList[Series[x^9/Product[1-t x,{t,9,18}],{x,0,30}],x] (* or *) LinearRecurrence[{135,-8160,290790,-6765213,107358615,-1176812090, 8797620060,-42924478536,123418922400, -158789030400}, {0,0,0,0,0,0,0,0,0,1},31] (* Harvey P. Dale, May 22 2012 *)

Formula

G.f.: x^9/ ((1-9x)(1-10x)(1-11x)(1-12x)(1-13x)(1-14x)(1-15x)(1-16x)(1-17x)(1-18*x)).

a(n)=0 for n<9, a(9)=1, a(n) = 135*a(n-1) -8160*a(n-2) +290790*a(n-3) -6765213*a(n-4) [phone]*a(n-5) -1176812090*a(n-6) [phone]*a(n-7) -42924478536*a(n-8) [phone]*a(n-9) -158789030400*a(n-10). - Harvey P. Dale, May 22 2012

E.g.f.: exp(9*x)*((exp(x)-1)^9)/9!. - Alois P. Heinz, May 04 2016