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
Examples
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;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
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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 *) -
PARI
T(n,k)=binomial(n*(n+1)*(n+2)/3!-k*(k+1)*(k+2)/3!, n-k)
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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
Formula
T(n,k) = C(n*(n+1)*(n+2)/3! - k*(k+1)*(k+2)/3!, n-k) for n >= k >= 0.
Comments