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A062160
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Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by falling antidiagonals.
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5
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0, 1, 0, -1, 1, 0, 1, 0, 1, 0, -1, 1, 1, 1, 0, 1, 0, 3, 2, 1, 0, -1, 1, 5, 7, 3, 1, 0, 1, 0, 11, 20, 13, 4, 1, 0, -1, 1, 21, 61, 51, 21, 5, 1, 0, 1, 0, 43, 182, 205, 104, 31, 6, 1, 0, -1, 1, 85, 547, 819, 521, 185, 43, 7, 1, 0, 1, 0, 171, 1640, 3277, 2604, 1111, 300, 57, 8, 1, 0, -1, 1, 341, 4921, 13107, 13021, 6665, 2101, 455, 73, 9, 1, 0
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OFFSET
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0,18
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LINKS
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FORMULA
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T(n, k) = n^(k-1)-n^(k-2)+n^(k-3)-...+(-1)^(k-1) = n^(k-1)-T(n, k-1) = nT(n, k-1)-(-1)^k = (n-1)T(n, k-1)+nT(n, k-2) = round[n^k/(n+1)] for n>1.
T(n,k) = (-1)^(k+1) * resultant( n*x + 1, (x^k-1)/(x-1) ). - Max Alekseyev, Sep 28 2021
E.g.f. of row n: (exp(n*x) - exp(-x))/(n+1). - Stefano Spezia, Feb 20 2024
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EXAMPLE
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Square array begins:
0, 1, -1, 1, -1, 1, -1, 1, ...
0, 1, 0, 1, 0, 1, 0, 1, ...
0, 1, 1, 3, 5, 11, 21, 43, ...
0, 1, 2, 7, 20, 61, 182, 547, ...
0, 1, 3, 13, 51, 205, 819, 3277, ...
0, 1, 4, 21, 104, 521, 2604, 13021, ...
0, 1, 5, 31, 185, 1111, 6665, 39991, ...
0, 1, 6, 43, 300, 2101, 14706, 102943, ... (End)
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MATHEMATICA
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T[n_, k_]:=(n^k - (-1)^k)/(n+1); Join[{0}, Table[Reverse[Table[T[n-k, k], {k, 0, n}]], {n, 12}]]//Flatten (* Stefano Spezia, Feb 20 2024 *)
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CROSSREFS
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Rows include A062157, A000035, A001045, A015518, A015521, A015531, A015540, A015552, A015565, A015577, A015585, A015592 etc.
Related to repunits in negative bases (cf. A055129 for positive bases).
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KEYWORD
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AUTHOR
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STATUS
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approved
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