A122175 Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 7, 1, 1, 5, 20, 28, 11, 1, 1, 6, 35, 84, 66, 16, 1, 1, 7, 56, 210, 286, 136, 22, 1, 1, 8, 84, 462, 1001, 816, 253, 29, 1, 1, 9, 120, 924, 3003, 3876, 2024, 435, 37, 1, 1, 10, 165, 1716, 8008, 15504, 12650, 4495, 703, 46, 1, 1, 11, 220

Offset: 0

Author

Paul D. Hanna, Aug 25 2006

Keywords

• nonn
• tabl

Comments

Remarkably, column k of the matrix inverse (A121435) equals signed column k of matrix power: A107876^(k*(k+1)/2 + 1).

Examples

			Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 4, 1;
1, 4, 10, 7, 1;
1, 5, 20, 28, 11, 1;
1, 6, 35, 84, 66, 16, 1;
1, 7, 56, 210, 286, 136, 22, 1;
1, 8, 84, 462, 1001, 816, 253, 29, 1; ...
		

Crossrefs

Cf. A121435 (inverse); variants: A098568, A122176, A122177.

Programs

Mathematica

Table[Binomial[(k(k+1))/2+n-k,n-k],{n,0,20},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 22 2016 *)

PARI

T(n,k)=if(n