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%I #11 Aug 29 2019 17:24:17
%S 1,600,600,648000,200,2592000,1270080000,25,2871000,13592880000,
%T 4267468800000,1,1294920,36462182400,100221504768000,
%U 23228686172160000,284800,38559024000,551224880640000,1056582600192000000
%N Array used for numerators of g.f.s for column sequences of array A090216 ((5,5)-Stirling2).
%C The row length sequence for this array is A090223(k-5)+1= floor(4*(k-5)/5)+1, k>=5: [1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, ...].
%C The g.f. G(k,x) for the k-th column (with leading zeros) of array A090216 is given there. The recurrence is G(k,x) = x*sum(binomial(k-r,5-r)*fallfac(5,5-r)*G(k-r,x),r=1..5))/(1-fallfac(k,5)*x), k>=5, with inputs G(k,x)=0 for k=1,2,3,4 and G(5,x)=x/(1-5!*x); where fallfac(n,m) := A008279(n,m) (falling factorials with fallfac(n,0) := 1). Computed from the Blasiak et al. reference, eqs. (20) and (21) with r=5: recurrence for S_{5,5}(n,k).
%H W. Lang, <a href="/A090222/a090222.txt">First 7 rows</a>.
%F a(k, n) from: sum(a(k, n)*x^n, n=0..kmax(k)) = G(k, x)* product(1-fallfac(p, 5)*x, p=5..k)/x^ceiling(k/5), k>=5, with G(k, x) defined from the recurrence given above and kmax(k) := floor(4*(k-5)/5)= A090223(k-5).
%e [1]; [600]; [648000,200]; [2592000,1270080000,25]; ...
%e G(6,x)/x^2 = 600/((1-5!*x)*(1-6*5*4*3*2*x)). kmax(6)=0, hence P(6,x)=a(6,0)=600; x^2 from x^ceiling(6/5).
%K nonn,easy,tabf
%O 5,2
%A _Wolfdieter Lang_, Dec 01 2003
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