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A322083 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+d)*d^k. 15
1, 1, -2, 1, -3, 2, 1, -5, 4, -1, 1, -9, 10, -3, 2, 1, -17, 28, -13, 6, -4, 1, -33, 82, -57, 26, -12, 2, 1, -65, 244, -241, 126, -50, 8, 0, 1, -129, 730, -993, 626, -252, 50, -3, 3, 1, -257, 2188, -4033, 3126, -1394, 344, -45, 13, -4, 1, -513, 6562, -16257, 15626, -8052, 2402, -441, 91, -18, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..66.

Index entries for sequences mentioned by Glaisher

FORMULA

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

EXAMPLE

Square array begins:

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

  -2,  -3,  -5,   -9,   -17,   -33,  ...

   2,   4,  10,   28,    82,   244,  ...

  -1,  -3, -13,  -57,  -241,  -993,  ...

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

  -4, -12, -50, -252, -1394, -8052,  ...

MATHEMATICA

Table[Function[k, Sum[(-1)^(n/d+d) d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

Table[Function[k, SeriesCoefficient[Sum[(-1)^(j + 1) j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

PROG

(PARI) T(n, k)={sumdiv(n, d, (-1)^(n/d+d)*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 A228441, A109506, A321558, A321559, A321560, A321561, A321562, A321563, A321564, A321565, A321807, A321808, A321809.

Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322084.

Sequence in context: A179314 A204927 A119441 * A058399 A209434 A207611

Adjacent sequences:  A322080 A322081 A322082 * A322084 A322085 A322086

KEYWORD

sign,tabl

AUTHOR

Ilya Gutkovskiy, Nov 26 2018

STATUS

approved

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Last modified May 29 20:42 EDT 2020. Contains 334710 sequences. (Running on oeis4.)