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A123963
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Triangle T(n, k) = k^4 - n^4 + 2*k*n*(1 - k^2*n^2), read by rows.
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1
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0, -1, 0, -16, -27, -120, -81, -128, -485, -1440, -256, -375, -1248, -3607, -8160, -625, -864, -2589, -7264, -16329, -31200, -1296, -1715, -4712, -12843, -28640, -54611, -93240, -2401, -3072, -7845, -20800, -45993, -87456, -149197, -235200, -4096, -5103, -12240, -31615, -69312, -131391, -223888, -352815, -524160
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OFFSET
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0,4
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COMMENTS
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A triangular sequence based on the omega(3) Jacobian Elliptic Modular equation.
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LINKS
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FORMULA
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T(n, k) = k^4 - n^4 + 2*k*n*(1 - k^2*n^2).
Sum_{k=0..n} T(n, k) = (-1/15)*binomial(n+1, 2) * (15*n^5 +15*n^4 +24*n^3 -9*n^2 -31*n +1). - G. C. Greubel, Feb 20 2021
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EXAMPLE
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Triangular sequence:
0;
-1, 0;
-16, -27, -120;
-81, -128, -485, -1440;
-256, -375, -1248, -3607, -8160;
-625, -864, -2589, -7264, -16329, -31200;
-1296, -1715, -4712, -12843, -28640, -54611, -93240;
-2401, -3072, -7845, -20800, -45993, -87456, -149197, -235200;
-4096, -5103, -12240, -31615, -69312, -131391, -223888, -352815, -524160;
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MATHEMATICA
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T[n_, k_]:= k^4 - n^4 + 2*n*k*(1 - k^2*n^2);
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 20 2021 *)
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PROG
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(Sage) flatten([[k^4 - n^4 + 2*n*k*(1 - k^2*n^2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 20 2021
(Magma) [k^4 - n^4 + 2*n*k*(1 - k^2*n^2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 20 2021
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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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