A126445 Triangle, read by rows, where T(n,k) = C(C(n+2,3) - C(k+2,3), n-k) for n >= k >= 0.

1, 1, 1, 6, 3, 1, 120, 36, 6, 1, 4845, 969, 120, 10, 1, 324632, 46376, 4495, 300, 15, 1, 32468436, 3478761, 270725, 15180, 630, 21, 1, 4529365776, 377447148, 24040016, 1150626, 41664, 1176, 28, 1, 840261910995, 56017460733, 2967205528, 122391522, 3921225, 98770, 2016, 36, 1

Offset

0,4

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

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

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

Example

Formula: T(n,k) = C(C(n+2,3) - C(k+2,3), n-k) is illustrated by:
T(n=4,k=1) = C(C(6,3) - C(3,3), n-k) = C(19,3) = 969;
T(n=4,k=2) = C(C(6,3) - C(4,3), n-k) = C(16,2) = 120;
T(n=5,k=2) = C(C(7,3) - C(4,3), n-k) = C(31,3) = 4495.
Triangle begins:
           1;
           1,         1;
           6,         3,        1;
         120,        36,        6,       1;
        4845,       969,      120,      10,     1;
      324632,     46376,     4495,     300,    15,    1;
    32468436,   3478761,   270725,   15180,   630,   21,  1;
  4529365776, 377447148, 24040016, 1150626, 41664, 1176, 28, 1;

Mathematica

T[n_, k_]:= Binomial[Binomial[n+2, 3] - Binomial[k+2, 3], n-k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2022 *)

Prog

(PARI) T(n, k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!, n-k)
(Sage)
def A126445(n, k): return binomial(binomial(n+2, 3) - binomial(k+2, 3), n-k)
flatten([[A126445(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 18 2022

Crossrefs

Columns: A126446, A126447, A126448, A126449 (row sums).

Variants: A107862, A126450, A126454, A126457.

Sequence in context: A182227 A108451 A122178 * A277435 A033326 A068996

Adjacent sequences: A126442 A126443 A126444 * A126446 A126447 A126448

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 27 2006

Status

approved