A121334 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k, n-k), for n>=k>=0.

1, 2, 1, 10, 4, 1, 84, 28, 7, 1, 1001, 286, 66, 11, 1, 15504, 3876, 816, 136, 16, 1, 296010, 65780, 12650, 2024, 253, 22, 1, 6724520, 1344904, 237336, 35960, 4495, 435, 29, 1, 177232627, 32224114, 5245786, 749398, 91390, 9139, 703, 37, 1, 5317936260

Offset

0,2

Comments

A triangle having similar properties and complementary construction is the dual triangle A122175.

Links

Table of n, a(n) for n=0..45.

Formula

Remarkably, row n of the matrix inverse (A121439) equals row n of A121412^(-n*(n+1)/2-1). Further, the following matrix products of triangles of binomial coefficients are equal: A121412 = A121334*A122178^-1 = A121335*A121334^-1 = A121336*A121335^-1, where row n of H=A121412 equals row (n-1) of H^(n+1) with an appended '1'.

Example

Triangle begins:
1;
2, 1;
10, 4, 1;
84, 28, 7, 1;
1001, 286, 66, 11, 1;
15504, 3876, 816, 136, 16, 1;
296010, 65780, 12650, 2024, 253, 22, 1;
6724520, 1344904, 237336, 35960, 4495, 435, 29, 1;
177232627, 32224114, 5245786, 749398, 91390, 9139, 703, 37, 1; ...

Prog

(PARI) T(n, k)=binomial(n*(n+1)/2+n-k, n-k)

Crossrefs

Cf. A121439 (matrix inverse); A121412; variants: A122178, A121335, A121336; A122175 (dual).

Sequence in context: A110327 A105615 A136216 * A126450 A235608 A112333

Adjacent sequences: A121331 A121332 A121333 * A121335 A121336 A121337

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 29 2006

Status

approved