A090222 Array used for numerators of g.f.s for column sequences of array A090216 ((5,5)-Stirling2).

1, 600, 600, 648000, 200, 2592000, 1270080000, 25, 2871000, 13592880000, 4267468800000, 1, 1294920, 36462182400, 100221504768000, 23228686172160000, 284800, 38559024000, 551224880640000, 1056582600192000000

Offset: 5

Author

Wolfdieter Lang, Dec 01 2003

Keywords

• nonn
• easy
• tabf

Comments

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, ...].

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).

Examples

			[1]; [600]; [648000,200]; [2592000,1270080000,25]; ...
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).
		

Links

• W. Lang, First 7 rows.

Formula

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).