A055403 -id:A055403 - 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.

A341423 Number of positive solutions to (x_1)^2 + (x_2)^2 + (x_3)^2 + (x_4)^2 <= n^2.

Original entry on oeis.org

1, 5, 32, 94, 219, 437, 804, 1362, 2177, 3271, 4768, 6708, 9227, 12381, 16254, 20954, 26707, 33461, 41480, 50884, 61703, 74183, 88606, 104862, 123481, 144241, 167604, 193648, 222799, 254731, 290244, 329512, 372545, 419661, 470822, 526646, 587481, 653505

Offset: 2

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Index entries for sequences related to sums of squares

Cf. A000122, A001182, A046895, A055403, A055410, A063730, A224213, A253663, A302995, A341424, A341425, A341426, A341427, A341428, A341429.

b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 4):
seq(a(n), n=2..39);  # Alois P. Heinz, Feb 11 2021

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^4/(16 (1 - x)), {x, 0, n^2}], {n, 2, 39}]

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^4 / (16 * (1 - x)).

A302863 a(n) = [x^(n^2)] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 6, 29, 165, 1203, 9763, 83877, 793049, 7903501, 83570177, 933697153, 10905583809, 133352809334, 1695473999478, 22354920990148, 305096197935075, 4296142551821184, 62336908825014452, 930284705538262688, 14255992611680074754, 224065160215526683317, 3607018540134004189466

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

a(n) = number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_n)^2 <= n^2.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Main diagonal of A302998.

Cf. A000603, A000604, A010052, A055403, A055404, A055405, A055406, A055407, A055408, A055409, A298329, A302861, A302862.

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/(2^n (1 - x)), {x, 0, n^2}], {n, 0, 22}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 0, n}]^n, {x, 0, n^2}], {n, 0, 22}]

A349611 Number of solutions to x^2 + y^2 + z^2 + w^2 <= n^2, where x, y, z, w are positive odd integers.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 32, 44, 82, 120, 207, 277, 405, 541, 768, 966, 1272, 1592, 2087, 2489, 3103, 3719, 4588, 5348, 6386, 7522, 8891, 10175, 11909, 13623, 15818, 17742, 20278, 22720, 25923, 28917, 32361, 36031, 40368, 44488, 49400, 54358, 60377, 65835, 72341

Offset: 0

Ilya Gutkovskiy, Nov 23 2021

nonn

			a(4) = 5 since there are solutions (1,1,1,1), (3,1,1,1), (1,3,1,1), (1,1,3,1), (1,1,1,3).

Robert Israel, Table of n, a(n) for n = 0..2500
Index entries for sequences related to sums of squares

Cf. A055403, A055410, A341423, A349609, A349610.

N:= 100: # for a(0) .. a(N)
F:= add(x^(k^2),k = 1 ... N,2):
F:= expand(F^4):
L:= ListTools:-PartialSums([seq](coeff(F,x,n),n=0..N^2)):
L[[seq(n^2+1,n=0..N)]]; # Robert Israel, Dec 21 2023

Table[SeriesCoefficient[EllipticTheta[2, 0, x^4]^4/(16 (1 - x)), {x, 0, n^2}], {n, 0, 44}]

a(n) = [x^(n^2)] theta_2(x^4)^4 / (16 * (1 - x)).

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A341423 Number of positive solutions to (x_1)^2 + (x_2)^2 + (x_3)^2 + (x_4)^2 <= n^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A302863 a(n) = [x^(n^2)] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A349611 Number of solutions to x^2 + y^2 + z^2 + w^2 <= n^2, where x, y, z, w are positive odd integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula