A322084 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.
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
Offset: 1
Examples
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, ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
- Index entries for sequences mentioned by Glaisher
Crossrefs
Programs
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Mathematica
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 -
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
Formula
G.f. of column k: Sum_{j>=1} j^k*x^j/(1 + x^(2*j)).