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A322084
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Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d==1 (mod 4)} d^k - Sum_{d|n, n/d==3 (mod 4)} d^k.
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5
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1, 1, 1, 1, 2, 0, 1, 4, 2, 1, 1, 8, 8, 4, 2, 1, 16, 26, 16, 6, 0, 1, 32, 80, 64, 26, 4, 0, 1, 64, 242, 256, 126, 32, 6, 1, 1, 128, 728, 1024, 626, 208, 48, 8, 1, 1, 256, 2186, 4096, 3126, 1280, 342, 64, 7, 2, 1, 512, 6560, 16384, 15626, 7744, 2400, 512, 73, 12, 0, 1, 1024, 19682, 65536, 78126, 46592, 16806, 4096, 703, 104, 10, 0
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OFFSET
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1,5
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LINKS
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FORMULA
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G.f. of column k: Sum_{j>=1} j^k*x^j/(1 + x^(2*j)).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 4, 8, 16, 32, ...
0, 2, 8, 26, 80, 242, ...
1, 4, 16, 64, 256, 1024, ...
2, 6, 26, 126, 626, 3126, ...
0, 4, 32, 208, 1280, 7744, ...
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MATHEMATICA
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Table[Function[k, SeriesCoefficient[Sum[j^k x^j/(1 + x^(2 j)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
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PROG
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(PARI) T(n, k)={sumdiv(n, d, if(d%2, (-1)^((d-1)/2)*(n/d)^k))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018
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CROSSREFS
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Columns k=0..12 give A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.
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KEYWORD
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AUTHOR
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STATUS
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approved
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