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

%I A143400 #17 Aug 30 2018 19:59:52
%S A143400 0,0,0,0,0,1,45,1190,24150,416451,6427575,91549480,1227283200,
%T A143400 15695180501,193333245105,2310273772170,26927270656650,
%U A143400 307413790470151,3449088814306635,38132767214613260,416342920938136500,4497187699884973401,48129773048982636165
%N A143400 Expansion of x^k/Product_{t=k..2k} (1-tx) for k=5.
%C A143400 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.
%H A143400 Alois P. Heinz, <a href="/A143400/b143400.txt">Table of n, a(n) for n = 0..300</a>
%H A143400 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A143400 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (45, -835, 8175, -44524, 127860, -151200).
%F A143400 G.f.: x^5/((1-5x)(1-6x)(1-7x)(1-8x)(1-9x)(1-10x)).
%F A143400 E.g.f.: exp(5*x)*((exp(x)-1)^5)/5!.
%p A143400 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);
%t A143400 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 *)
%Y A143400 5th column of A143395.
%K A143400 nonn
%O A143400 0,7
%A A143400 _Alois P. Heinz_, Aug 12 2008