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.

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,9

LINKS

Table of n, a(n) for n=1..78.

Index entries for sequences mentioned by Glaisher

FORMULA

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

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, ...

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

1, 1, 1, 1, 1, 1, ...

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

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

MATHEMATICA

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

CROSSREFS

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

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

Sequence in context: A329042 A171660 A157117 * A264081 A061538 A123602

Adjacent sequences: A322140 A322141 A322142 * A322144 A322145 A322146

KEYWORD

sign,tabl

AUTHOR

Ilya Gutkovskiy, Nov 28 2018

STATUS

approved