A122177 Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k + 2, n-k) for n>=k>=0.

1, 3, 1, 6, 4, 1, 10, 10, 6, 1, 15, 20, 21, 9, 1, 21, 35, 56, 45, 13, 1, 28, 56, 126, 165, 91, 18, 1, 36, 84, 252, 495, 455, 171, 24, 1, 45, 120, 462, 1287, 1820, 1140, 300, 31, 1, 55, 165, 792, 3003, 6188, 5985, 2600, 496, 39, 1, 66, 220, 1287, 6435, 18564, 26334

Offset

0,2

Comments

Remarkably, column k of the matrix inverse (A121437) equals signed column k of matrix power: A107876^(k*(k+1)/2 + 3) for k>=0.

Links

Table of n, a(n) for n=0..60.

Example

Triangle begins:
1;
3, 1;
6, 4, 1;
10, 10, 6, 1;
15, 20, 21, 9, 1;
21, 35, 56, 45, 13, 1;
28, 56, 126, 165, 91, 18, 1;
36, 84, 252, 495, 455, 171, 24, 1;
45, 120, 462, 1287, 1820, 1140, 300, 31, 1; ...

Mathematica

A122177[n_, k_] := Binomial[k*(k + 1)/2 + n - k + 2, n - k];
Table[A122177[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jul 23 2024 *)

Prog

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

Crossrefs

Cf. A121437 (inverse); variants: A098568, A122175, A122176.

Sequence in context: A086270 A325000 A104712 * A255874 A108286 A185944

Adjacent sequences: A122174 A122175 A122176 * A122178 A122179 A122180

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 25 2006

Status

approved