A089517 Array used for numerators of g.f.s for column sequences of array A078741 ((3,3)-Stirling2).

1, 18, 9, 432, 1, 672, 14400, 243, 47520, 648000, 27, 36396, 3790800, 38102400, 1, 9765, 5115888, 354715200, 2844979200, 1107, 2499552, 757646784, 39182330880, 263363788800, 54, 546453, 592216272, 123294623040, 5089348454400

Offset: 3

Author

Wolfdieter Lang, Dec 01 2003

Keywords

• nonn
• easy
• tabf

Comments

The row length sequence for this array is A004396(n-2)=floor((2*n-3)/3), n>=3: [1,1,2,3,3,4,5,5,6,7,7,8,9,9,10,...].

The g.f. G(m,x) for the m-th column sequence (with leading zeros) of array A078741 is given there. The recurrence is G(m,x) = x*(3*fallfac(m-1,2)*G(m-1,x) + 3*(m-2)*G(m-2,x) + G(m-3,x))/(1-fallfac(m,3)*x), m>=4, with inputs G(1,x)=0=G(2,x) and G(3,x)=x/(1-(3*2*1)*x); where fallfac(n,m) := A008279(n,m) (falling factorials). Computed from the Blasiak et al. reference, eqs. (20) and (21) with r=3: recurrence for S_{3,3}(n,k).

Examples

			[1]; [18]; [9,423]; [1,672,14400]; [243,47520,648000]; ...
G(4,x)/(x^2) = 18/((1-3*2*1*x)*(1-4*3*2*x)). kmax(4)=0, hence P(4,x)=a(4,0)=18; x^2 from x^ceiling(4/3).
		

Links

• W. Lang, First 11 rows.

Formula

a(n, m) from: sum(a(n, m)*x^m, m=0..kmax(n)) = G(n, x)* product(1-fallfac(p, 3)*x, p=3..n)/x^ceiling(n/3), n>=3, with G(n, x) defined from the recurrence given above and kmax(n) := A004523(n-3)= floor(2*(n-3)/3) = A004396(n-3)-1.