A121335 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 1, n-k), for n>=k>=0.

1, 3, 1, 15, 5, 1, 120, 36, 8, 1, 1365, 364, 78, 12, 1, 20349, 4845, 969, 153, 17, 1, 376740, 80730, 14950, 2300, 276, 23, 1, 8347680, 1623160, 278256, 40920, 4960, 465, 30, 1, 215553195, 38320568, 6096454, 850668, 101270, 9880, 741, 38, 1, 6358402050

Offset

0,2

Comments

A triangle having similar properties and complementary construction is the dual triangle A122176.

Links

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

Formula

Remarkably, row n of the matrix inverse (A121440) equals row n of A121412^(-n*(n+1)/2-2). 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;
3, 1;
15, 5, 1;
120, 36, 8, 1;
1365, 364, 78, 12, 1;
20349, 4845, 969, 153, 17, 1;
376740, 80730, 14950, 2300, 276, 23, 1;
8347680, 1623160, 278256, 40920, 4960, 465, 30, 1;
215553195, 38320568, 6096454, 850668, 101270, 9880, 741, 38, 1; ...

Prog

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

Crossrefs

Cf. A121440 (matrix inverse); A121412; variants: A122178, A121334, A121336; A122176 (dual).

Sequence in context: A190088 A119301 A293157 * A126454 A293558 A259841

Adjacent sequences: A121332 A121333 A121334 * A121336 A121337 A121338

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 29 2006

Status

approved