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A302998 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] (1 + theta_3(x))^k/(2^k*(1 - x)), where theta_3() is the Jacobi theta function. 3
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 11, 11, 5, 1, 1, 6, 20, 29, 17, 6, 1, 1, 7, 36, 70, 54, 26, 7, 1, 1, 8, 63, 157, 165, 99, 35, 8, 1, 1, 9, 106, 337, 482, 357, 163, 45, 9, 1, 1, 10, 171, 702, 1319, 1203, 688, 239, 58, 10, 1, 1, 11, 265, 1420, 3390, 3819, 2673, 1154, 344, 73, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

A(n,k) is the number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_k)^2 <= n^2.

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>=0} x^(j^2))^k.

EXAMPLE

Square array begins:

1,  1,   1,   1,    1,     1,  ...

1,  2,   3,   4,    5,     6,  ...

1,  3,   6,  11,   20,    36,  ...

1,  4,  11,  29,   70,   157,  ...

1,  5,  17,  54,  165,   482,  ...

1,  6,  26,  99,  357,  1203,  ...

MATHEMATICA

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

CROSSREFS

Columns k=0..10 give A000012, A000027, A000603, A000604, A055403, A055404, A055405, A055406, A055407, A055408, A055409.

Main diagonal gives A302863.

Cf. A000122, A122510, A302996, A302997.

Sequence in context: A230861 A119724 A162424 * A303484 A008571 A230860

Adjacent sequences:  A302995 A302996 A302997 * A302999 A303000 A303001

KEYWORD

nonn,tabl

AUTHOR

Ilya Gutkovskiy, Apr 17 2018

STATUS

approved

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Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)