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

0,5

Comments

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

Links

Table of n, a(n) for n=0..68.

Example

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; ...

Mathematica

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

Prog

(PARI) T(n, k)=if(n<k|k<0, 0, binomial(k*(k+1)/2+n-k, n-k))

Crossrefs

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

Sequence in context: A098447 A175105 A162717 * A073165 A137153 A340814

Adjacent sequences: A122172 A122173 A122174 * A122176 A122177 A122178

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 25 2006

Status

approved