A122178 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k - 1, n-k), for n>=k>=0.

1, 1, 1, 6, 3, 1, 56, 21, 6, 1, 715, 220, 55, 10, 1, 11628, 3060, 680, 120, 15, 1, 230230, 53130, 10626, 1771, 231, 21, 1, 5379616, 1107568, 201376, 31465, 4060, 406, 28, 1, 145008513, 26978328, 4496388, 658008, 82251, 8436, 666, 36, 1, 4431613550

Offset

0,4

Comments

A triangle having similar properties and complementary construction is the dual triangle A098568.

Links

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

Formula

Remarkably, row n of the matrix inverse (A121438) equals row n of A121412^(-n*(n+1)/2). 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;
1, 1;
6, 3, 1;
56, 21, 6, 1;
715, 220, 55, 10, 1;
11628, 3060, 680, 120, 15, 1;
230230, 53130, 10626, 1771, 231, 21, 1;
5379616, 1107568, 201376, 31465, 4060, 406, 28, 1;
145008513, 26978328, 4496388, 658008, 82251, 8436, 666, 36, 1; ...

Prog

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

Crossrefs

Cf. A121438 (matrix inverse); A121412; variants: A121334, A121335, A121336; A098568 (dual).

Sequence in context: A243424 A182227 A108451 * A126445 A277435 A033326

Adjacent sequences: A122175 A122176 A122177 * A122179 A122180 A122181

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 29 2006

Status

approved