A322143 - OEIS

A322143

Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, d==1 (mod 4)} d^k - Sum_{d|n, d==3 (mod 4)} d^k.

%I #4 Dec 01 2018 09:18:50

%S 1,1,1,1,1,0,1,1,-2,1,1,1,-8,1,2,1,1,-26,1,6,0,1,1,-80,1,26,-2,0,1,1,

%T -242,1,126,-8,-6,1,1,1,-728,1,626,-26,-48,1,1,1,1,-2186,1,3126,-80,

%U -342,1,7,2,1,1,-6560,1,15626,-242,-2400,1,73,6,0,1,1,-19682,1,78126,-728,-16806,1,703,26,-10,0

%N Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, d==1 (mod 4)} d^k - Sum_{d|n, d==3 (mod 4)} d^k.

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>

%F G.f. of column k: Sum_{j>=1} (-1)^(j-1)*(2*j - 1)^k*x^(2*j-1)/(1 - x^(2*j-1)).

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 0, -2, -8, -26, -80, -242, ...

%e 1, 1, 1, 1, 1, 1, ...

%e 2, 6, 26, 126, 626, 3126, ...

%e 0, -2, -8, -26, -80, -242, ...

%t Table[Function[k, SeriesCoefficient[Sum[(-1)^(j - 1) (2 j - 1)^k x^(2 j - 1)/(1 - x^(2 j - 1)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

%Y Columns k=0..12 give A002654, A050457, A002173, A050459, A050456, A321821, A321822, A321823, A321824, A321825, A321826, A321827, A321828.

%Y Cf. A109974, A322081, A322082, A322083, A322084.

%K sign,tabl

%O 1,9

%A _Ilya Gutkovskiy_, Nov 28 2018