A157634 Triangle T(n, k) = 1 if k = 0 or k = n, otherwise n^5 - k^5 - (n-k)^5, read by rows.
1, 1, 1, 1, 30, 1, 1, 210, 210, 1, 1, 780, 960, 780, 1, 1, 2100, 2850, 2850, 2100, 1, 1, 4650, 6720, 7290, 6720, 4650, 1, 1, 9030, 13650, 15540, 15540, 13650, 9030, 1, 1, 15960, 24960, 29400, 30720, 29400, 24960, 15960, 1, 1, 26280, 42210, 51030, 54900, 54900, 51030, 42210, 26280, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 30, 1; 1, 210, 210, 1; 1, 780, 960, 780, 1; 1, 2100, 2850, 2850, 2100, 1; 1, 4650, 6720, 7290, 6720, 4650, 1; 1, 9030, 13650, 15540, 15540, 13650, 9030, 1; 1, 15960, 24960, 29400, 30720, 29400, 24960, 15960, 1; 1, 26280, 42210, 51030, 54900, 54900, 51030, 42210, 26280, 1; 1, 40950, 67200, 82950, 91200, 93750, 91200, 82950, 67200, 40950, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Magma
A157634:= func< n,k | k eq 0 or k eq n select 1 else n^5 - (k^5 + (n-k)^5) >; [A157634(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 13 2021
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Mathematica
T[n_, k_]:= If[n*k*(n-k)==0, 1, n^5 - (k^5 + (n-k)^5)]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten -
Sage
def A157634(n,k): return 1 if (k==0 or k==n) else n^5 - (k^5 + (n-k)^5) flatten([[A157634(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 13 2021
Formula
T(n, k) = 1 if k = 0 or k = n, otherwise 5*n*k*(n-k)*(n^2 -n*k +k^2).
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = 2 - [n=0] + 30*A006858(n).
From G. C. Greubel, Dec 13 2021: (Start)
T(n, 1) = [n<2] + 30*A006325(n).
T(2*n, n) = [n=0] + 30*A000584(n). (End)