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1, 1, 1, 1, 8, 1, 1, 27, 27, 1, 1, 64, 83, 64, 1, 1, 125, 181, 181, 125, 1, 1, 216, 333, 370, 333, 216, 1, 1, 343, 551, 649, 649, 551, 343, 1, 1, 512, 847, 1036, 1097, 1036, 847, 512, 1, 1, 729, 1233, 1549, 1701, 1701, 1549, 1233, 729, 1, 1, 1000, 1721, 2206, 2485, 2576, 2485, 2206, 1721, 1000, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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COMMENTS
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Like A176282 but build on sums of cubes (A000537) instead of sums of squares.
Row sums are {1, 2, 10, 56, 213, 614, 1470, 3088, 5889, 10426, 17402, ...} = (n+1)*(9*n^4 + 6*n^3 - 11*n^2 - 4*n + 60)/60.
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LINKS
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FORMULA
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T(n,k) = T(n,n-k).
T(n, k) = (4 + n^2*(n+1)^2 - k^2*(k+1)^2 - (n-k)^2*(n-k+1)^2)/4. - G. C. Greubel, Nov 25 2019
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 8, 1;
1, 27, 27, 1;
1, 64, 83, 64, 1;
1, 125, 181, 181, 125, 1;
1, 216, 333, 370, 333, 216, 1;
1, 343, 551, 649, 649, 551, 343, 1;
1, 512, 847, 1036, 1097, 1036, 847, 512, 1;
1, 729, 1233, 1549, 1701, 1701, 1549, 1233, 729, 1;
1, 1000, 1721, 2206, 2485, 2576, 2485, 2206, 1721, 1000, 1;
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MAPLE
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MATHEMATICA
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(* sequences with q=1..10 *)
f[n_, k_, q_]:= f[n, k, q] = 1 + Sum[i^q, {i, 0, n}] - Sum[i^q, {i, 0, k}] + Sum[i^q, {i, 0, n-k}])); Table[Flatten[Table[f[n, k, q], {n, 0, 10}, {k, 0, n}]], {q, 1, 10}]
(* Second program *)
Table[(4 +n^2*(n+1)^2 -k^2*(k+1)^2 -(n-k)^2*(n-k+1)^2)/4, {n, 0, 12}, {k, 0, n} ]//Flatten (* G. C. Greubel, Nov 25 2019 *)
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PROG
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(PARI) T(n, k) = 1 + (n^2*(n+1)^2 - k^2*(k+1)^2 - (n-k)^2*(n-k+1)^2)/4; \\ G. C. Greubel, Nov 25 2019
(Magma) [(4 +n^2*(n+1)^2 -k^2*(k+1)^2 -(n-k)^2*(n-k+1)^2)/4: k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 25 2019
(Sage) [[(4 +n^2*(n+1)^2 -k^2*(k+1)^2 -(n-k)^2*(n-k+1)^2)/4 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 25 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> (4 +n^2*(n+1)^2 -k^2*(k+1)^2 - (n-k)^2*(n-k+1)^2)/4 ))); # G. C. Greubel, Nov 25 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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