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%I #15 Aug 09 2020 17:15:03
%S 0,0,0,0,0,0,720,12600,134400,1134000,8341200,56133000,355291200,
%T 2151864000,12614281680,72135063000,404672486400,2236228722000,
%U 12209943566160,66024457842600,354214283304000,1887999348060000,10008933180578640,52820388477271800,277680637970208000
%N a(n) = 5!*n*Stirling2(n-1, 5).
%C Original name: a simple grammar.
%H Andrew Howroyd, <a href="/A052785/b052785.txt">Table of n, a(n) for n = 0..200</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=742">Encyclopedia of Combinatorial Structures 742</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (30,-395,3000,-14523,46710,-100805,143700,-129076,65760,-14400).
%F E.g.f.: x*exp(x)^5-5*x*exp(x)^4+10*exp(x)^3*x-10*exp(x)^2*x+5*x*exp(x)-x.
%F Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, ( - 14400 - 27000*n^2 - 32880*n - 120*n^5 - 1800*n^4 - 10200*n^3)*a(n) + (42196*n^2 + 19454*n^3 + 32880*n + 274*n^5 + 3836*n^4)*a(n + 1) + ( - 13500*n - 13275*n^3 - 24075*n^2 - 225*n^5 - 2925*n^4)*a(n + 2) + (85*n^5 + 3400*n + 1020*n^4 + 4165*n^3 + 6630*n^2)*a(n + 3) + ( - 915*n^2 - 450*n - 615*n^3 - 15*n^5 - 165*n^4)*a(n + 4) + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a(n + 5)}.
%F From _Andrew Howroyd_, Aug 08 2020: (Start)
%F a(n) = n*A001118(n-1) for n > 1.
%F G.f.: 120*x^6*(2 - 5*x)*(3 - 30*x + 95*x^2 - 100*x^3 + 24*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x))^2.
%F E.g.f.: x*(exp(x) - 1)^5. (End)
%p spec := [S,{B=Set(Z,1 <= card),S=Prod(Z,B,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%o (PARI) a(n)={if(n>=1, 5!*n*stirling(n-1, 5, 2), 0)} \\ _Andrew Howroyd_, Aug 08 2020
%Y Cf. A001118, A052749.
%K easy,nonn
%O 0,7
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E Name changed and terms a(21) and beyond from _Andrew Howroyd_, Aug 08 2020
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