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

1, 96, 72, 14400, 16, 38400, 3456000, 1, 27000, 22104000, 1270080000, 7200, 34905600, 16111872000, 682795008000, 856, 21154176, 48248363520, 15279164006400, 516193026048000, 48, 6064128, 54644474880, 78083415244800

Offset: 4

Author

Wolfdieter Lang, Dec 01 2003

Keywords

• nonn
• easy
• tabf

Comments

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, ...].

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).

Examples

			[1]; [96]; [72,14400]; [16,38400,3456000]; [1,27000,22104000,1270080000]; ...
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).
		

Links

• W. Lang, First 8 rows.

Formula

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.