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 A126445 Triangle, read by rows, where T(n,k) = C(C(n+2,3) - C(k+2,3), n-k) for n >= k >= 0. 10
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified July 2 18:27 EDT 2022. Contains 355029 sequences. (Running on oeis4.)