A126454 Triangle, read by rows, where T(n,k) = C( C(n+2,3) - C(k+2,3) + 2, n-k) for n>=k>=0.

1, 3, 1, 15, 5, 1, 220, 55, 8, 1, 7315, 1330, 153, 12, 1, 435897, 58905, 5456, 351, 17, 1, 40475358, 4187106, 316251, 17296, 703, 23, 1, 5373200880, 437353560, 27285336, 1282975, 45760, 1275, 30, 1, 962889794295, 63140314380, 3295144749, 134153712

Offset

0,2

Comments

Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1.

Links

Table of n, a(n) for n=0..39.

Formula

T(n,k) = C( n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3! + 2, n-k) for n>=k>=0.

Example

Formula: T(n,k) = C( C(n+2,3) - C(k+2,3) + 2, n-k) is illustrated by:
T(n=4,k=1) = C( C(6,3) - C(3,3) + 2, n-k) = C(21,3) = 1330;
T(n=4,k=2) = C( C(6,3) - C(4,3) + 2, n-k) = C(18,2) = 153;
T(n=5,k=2) = C( C(7,3) - C(4,3) + 2, n-k) = C(33,3) = 5456.
Triangle begins:
1;
3, 1;
15, 5, 1;
220, 55, 8, 1;
7315, 1330, 153, 12, 1;
435897, 58905, 5456, 351, 17, 1;
40475358, 4187106, 316251, 17296, 703, 23, 1;
5373200880, 437353560, 27285336, 1282975, 45760, 1275, 30, 1; ...

Mathematica

Table[Binomial[Binomial[n+2, 3]-Binomial[k+2, 3]+2, n-k], {n, 0, 10}, {k, 0, n}]// Flatten (* Harvey P. Dale, Dec 17 2020 *)

Prog

(PARI) T(n, k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!+2, n-k)

Crossrefs

Columns: A126455, A126456; variants: A126445, A126450, A126457, A107870.

Sequence in context: A119301 A293157 A121335 * A293558 A259841 A228540

Adjacent sequences: A126451 A126452 A126453 * A126455 A126456 A126457

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 27 2006

Status

approved