%I #24 May 14 2016 14:49:28
%S 0,0,0,0,1,30,545,7770,95781,1071630,11192665,111095490,1060634861,
%T 9822843030,88799732385,787259974410,6869327386741,59158464019230,
%U 503954741177705,4254156112792530,35637875826743421,296621138907400230,2455329298857576625
%N Expansion of x^k/Product_{t=k..2k} (1-tx) for k=4.
%C 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.
%C 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
%H Alois P. Heinz, <a href="/A143399/b143399.txt">Table of n, a(n) for n = 0..350</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (30, -355, 2070, -5944, 6720).
%F G.f.: x^4/((1-4x)(1-5x)(1-6x)(1-7x)(1-8x)).
%F a(n) = 30a(n-1) -355a(n-2) +2070a(n-3) -5944a(n-4) +6720a(n-5).
%F E.g.f.: exp(4*x)*((exp(x)-1)^4)/4!. - _Wolfdieter Lang_, Oct 08 2011
%p 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);
%t LinearRecurrence[{30,-355,2070,-5944,6720},{0,0,0,0,1},30] (* _Harvey P. Dale_, Mar 12 2013 *)
%Y 4th column of A143395.
%K nonn
%O 0,6
%A _Alois P. Heinz_, Aug 12 2008