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A090221 Array used for numerators of g.f.s for column sequences of array A090214 ((4,4)-Stirling2).

%I A090221 #11 Aug 29 2019 17:59:11
%S A090221 1,96,72,14400,16,38400,3456000,1,27000,22104000,1270080000,7200,
%T A090221 34905600,16111872000,682795008000,856,21154176,48248363520,
%U A090221 15279164006400,516193026048000,48,6064128,54644474880,78083415244800
%N A090221 Array used for numerators of g.f.s for column sequences of array A090214 ((4,4)-Stirling2).
%C A090221 The row length sequence for this array is A037915(k-4)= floor(3*(k-4)/4)+1, k>=4: [1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, ...].
%C A090221 The g.f. G(k,x) for the k-th column (with leading zeros) of array A090214 is given there. The recurrence is G(k,x) = x*sum(binomial(k-r,4-r)*fallfac(4,4-r)*G(k-r,x),r=1..4))/(1-fallfac(k,4)*x), k>=4, with inputs G(k,x)=0 for k=1,2,3 and G(4,x)=x/(1-4!*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=4: recurrence for S_{4,4}(n,k).
%H A090221 W. Lang, <a href="/A090221/a090221.txt">First 8 rows</a>.
%F A090221 a(k, n) from: sum(a(k, n)*x^n, n=0..kmax(k)) = G(k, x)* product(1-fallfac(p, 4)*x, p=4..k)/x^ceiling(k/4), k>=4, with G(k, x) defined from the recurrence given above and kmax(k) := A057353(k-4)= floor(3*(k-4)/4)= A037915(k-4)-1.
%e A090221 [1]; [96]; [72,14400]; [16,38400,3456000]; [1,27000,22104000,1270080000]; ...
%e A090221 G(5,x)/x^2 = 96/((1-4!*x)*(1-5*4*3*2*x)). kmax(5)=0, hence P(5,x)=a(5,0)=96; x^2 from x^ceiling(5/4).
%K A090221 nonn,easy,tabf
%O A090221 4,2
%A A090221 _Wolfdieter Lang_, Dec 01 2003