A093447 Triangle a(n,k) read by rows n which contain columns k=1,2,..,n, where each entry is the product of numbers (k-1)*n-T(k-2)+1 through k*n-T(k-1).

1, 2, 3, 6, 20, 6, 24, 210, 72, 10, 120, 3024, 1320, 182, 15, 720, 55440, 32760, 4896, 380, 21, 5040, 1235520, 1028160, 175560, 13800, 702, 28, 40320, 32432400, 39070080, 7893600, 657720, 32736, 1190, 36, 362880, 980179200, 1744364160

Offset: 1

Author

Amarnath Murthy, Apr 02 2004

Keywords

• nonn
• tabl

Comments

This is built by starting from the sequence 1,2,....,T(n) in row n, where T(n) is the triangular number A000217(n) and packaging its first n, the next n-1, the next n-2,... up to the last number in groups and writing down the product of each group in one cell of the triangle. The first column is A000142. The second column is essentially A006963. The 3rd column is essentially A001763. The diagonal is A000217. - R. J. Mathar, Jul 26 2007

Examples

			In factorized notation the triangle starts
1;
1*2, 3;
1*2*3, 4*5, 6;
1*2*3*4, 5*6*7, 8*9, 10;
1*2*3*4*5, 6*7*8*9, 10*11*12, 13*14, 15;
which gives
1;
2, 3;
6, 20, 6;
24, 210, 72, 10;
120, 3024, 1320, 182, 15;
720,55440,32760, 4896, 380, 21;
		

Crossrefs

Cf. A093445, A093446, A093448.

Programs

Maple

A000217 := proc(n) n*(n+1)/2 ; end: A093447 := proc(n,k) factorial(k*n-A000217(k-1))/factorial((k-1)*n-A000217(k-2)) ; end: for n from 1 to 16 do for k from 1 to n do printf("%d, ",A093447(n,k)) ; od ; od: # R. J. Mathar, Jul 26 2007

Formula

a(n,k)= [k*n-T(k-1)]!/[(k-1)*n-T(k-2)]! where T(n)=A000217(n). - R. J. Mathar, Jul 26 2007

Extensions

More terms from R. J. Mathar, Jul 26 2007