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A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function. 2
1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 33, 29, 9, 1, 1, 11, 89, 123, 49, 11, 1, 1, 13, 221, 425, 257, 81, 13, 1, 1, 15, 485, 1343, 1281, 515, 113, 15, 1, 1, 17, 953, 4197, 5913, 3121, 925, 149, 17, 1, 1, 19, 1713, 12435, 23793, 16875, 6577, 1419, 197, 19, 1, 1, 21, 2869, 33809, 88273, 84769, 42205, 11833, 2109, 253, 21, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

A(n,k) is the number of integer lattice points inside the k-dimensional hypersphere of radius n.

LINKS

Table of n, a(n) for n=0..77.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Index entries for sequences related to sums of squares

FORMULA

A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j=-infinity..infinity} x^(j^2))^k.

EXAMPLE

Square array begins:

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

1,   3,   5,    7,     9,     11,  ...

1,   5,  13,   33,    89,    221,  ...

1,   7,  29,  123,   425,   1343,  ...

1,   9,  49,  257,  1281,   5913,  ...

1,  11,  81,  515,  3121,  16875,  ...

MATHEMATICA

Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k/(1 - x), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

CROSSREFS

Columns k=0..10 give A000012, A005408, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416.

Main diagonal gives A302861.

Cf. A000122, A122510, A302996, A302998.

Sequence in context: A026714 A008288 A238339 * A144461 A106597 A108359

Adjacent sequences:  A302994 A302995 A302996 * A302998 A302999 A303000

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Apr 17 2018

STATUS

approved

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Last modified January 22 05:11 EST 2019. Contains 319353 sequences. (Running on oeis4.)