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

%I A143404 #21 May 14 2016 14:52:52
%S A143404 0,0,0,0,0,0,0,0,0,1,135,10065,547965,24336312,934863930,32189799070,
%T A143404 1017281878470,30001945084683,835898091070185,22206607023852615,
%U A143404 566594907018764715,13964270139973201114,333991935681805199700,7781827783346875932300
%N A143404 Expansion of x^k/Product_{t=k..2k} (1-tx) for k=9.
%C A143404 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.
%H A143404 Alois P. Heinz, <a href="/A143404/b143404.txt">Table of n, a(n) for n = 0..300</a>
%H A143404 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A143404 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (135, -8160, 290790, -6765213, 107358615, -1176812090, 8797620060, -42924478536, 123418922400, -158789030400).
%F A143404 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)).
%F A143404 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) +107358615*a(n-5) -1176812090*a(n-6) +8797620060*a(n-7) -42924478536*a(n-8) +123418922400*a(n-9) -158789030400*a(n-10). - _Harvey P. Dale_, May 22 2012
%F A143404 E.g.f.: exp(9*x)*((exp(x)-1)^9)/9!. - _Alois P. Heinz_, May 04 2016
%p A143404 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);
%t A143404 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 *)
%Y A143404 9th column of A143395.
%K A143404 nonn
%O A143404 0,11
%A A143404 _Alois P. Heinz_, Aug 12 2008