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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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} j^k*x^j/(1 + x^(2*j)).

EXAMPLE

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,  ...

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

PROG

(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

CROSSREFS

Columns k=0..12 give A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322083.

Sequence in context: A289522 A316273 A124915 * A158239 A159819 A030188

Adjacent sequences:  A322081 A322082 A322083 * A322085 A322086 A322087

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Nov 26 2018

STATUS

approved

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Last modified July 14 13:27 EDT 2020. Contains 335729 sequences. (Running on oeis4.)