A055417 -id:A055417 - cp's OEIS frontend

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.

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

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

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

			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,  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Columns k=0..10 give A000012, A000027, A000603, A000604, A055403, A055404, A055405, A055406, A055407, A055408, A055409.
Rows n=0..10 give A000012, A000027, A055417, A055418, A055419, A055420, A055421, A055422, A055423, A055424, A055425.
Main diagonal gives A302863.
Cf. A000122, A122510, A302996, A302997.

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

T(n,k)={if(k==0, 1, polcoef(((sum(j=0, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019

A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j>=0} x^(j^2))^k.

A055418 Number of points in N^n of norm <= 3.

Original entry on oeis.org

1, 4, 11, 29, 70, 157, 337, 702, 1420, 2780, 5258, 9615, 17043, 29381, 49430, 81404, 131563, 209084, 327237, 504945, 768820, 1155781, 1716375, 2518938, 3654750, 5244356, 7445244, 10461091, 14552809, 20051645, 27374612, 37042552, 49701157

Offset: 0

David W. Wilson

			There are exactly 19 coordinate configurations (up to permutation) with up to 9 nonzero positive coordinates that can produce a vector of norm <= 3:
{..., 0, 0, 0, 0, 0, 0, 0, 0, 0}   0
{..., 0, 0, 0, 0, 0, 0, 0, 0, 1}   1
{..., 0, 0, 0, 0, 0, 0, 0, 0, 2}   2
{..., 0, 0, 0, 0, 0, 0, 0, 0, 3}   3
{..., 0, 0, 0, 0, 0, 0, 0, 1, 1}   sqrt(2)
{..., 0, 0, 0, 0, 0, 0, 0, 1, 2}   sqrt(5)
{..., 0, 0, 0, 0, 0, 0, 0, 2, 2}   2 sqrt(2)
{..., 0, 0, 0, 0, 0, 0, 1, 1, 1}   sqrt(3)
{..., 0, 0, 0, 0, 0, 0, 1, 1, 2}   sqrt(2) sqrt(3)
{..., 0, 0, 0, 0, 0, 0, 1, 2, 2}   3
{..., 0, 0, 0, 0, 0, 1, 1, 1, 1}   2
{..., 0, 0, 0, 0, 0, 1, 1, 1, 2}   sqrt(7)
{..., 0, 0, 0, 0, 1, 1, 1, 1, 1}   sqrt(5)
{..., 0, 0, 0, 0, 1, 1, 1, 1, 2}   2 sqrt(2)
{..., 0, 0, 0, 1, 1, 1, 1, 1, 1}   sqrt(6)
{..., 0, 0, 0, 1, 1, 1, 1, 1, 2}   3
{..., 0, 0, 1, 1, 1, 1, 1, 1, 1}   sqrt(7)
{..., 0, 1, 1, 1, 1, 1, 1, 1, 1}   2 sqrt(2)
{..., 1, 1, 1, 1, 1, 1, 1, 1, 1}   3
To produce the formula for a(n), it is sufficient to sum the number of permutations of these configurations in a vector of arbitrary length n.
This gives in the same order:
a(n) = 1 + n + n + n + binomial(n, 2) + n*(n - 1) + binomial(n, 2) + binomial(n, 3) + n*binomial(n-1, 2) + n*binomial(n-1, 2) + binomial(n, 4) + n*binomial(n-1, 3) + binomial(n, 5) + n*binomial(n-1, 4) + binomial(n, 6) + n*binomial(n-1, 5) + binomial(n, 7) + binomial(n, 8) + binomial(n, 9).
This is a polynomial of degree 9 in n.
a(n) = (1 + n) (9! + n (452016 + n (-224244 + n (152108 + n (-17351 + n (-16 + n (394 + (-28 + n) n)))))))/(9!).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Row n=3 of A302998.

Cf. A055417 (case for norm <= 2).

Satisfies a degree nine polynomial (see Example section). - Olivier Gérard, Mar 30 2015

G.f.: -(8*x^8-35*x^7+51*x^6-30*x^5-5*x^4+21*x^3-16*x^2+6*x-1) / (x-1)^10. - Colin Barker, Jul 07 2013

cp's OEIS Frontend

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.

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