A055404 -id:A055404 - 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.

A341424 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n^2.

Original entry on oeis.org

6, 51, 177, 547, 1348, 2958, 5574, 10084, 16974, 27450, 41970, 62671, 90216, 128082, 175867, 238018, 316373, 414998, 534094, 682144, 859705, 1075165, 1326551, 1627896, 1976582, 2390057, 2862607, 3411273, 4039483, 4760419, 5571729, 6500650, 7541560, 8722096

Offset: 3

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Index entries for sequences related to sums of squares

Cf. A000122, A001182, A055404, A055411, A175360, A253663, A302995, A340481, A341400, A341423, 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, 5):
seq(a(n), n=3..36);  # Alois P. Heinz, Feb 11 2021

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^5/(32 (1 - x)), {x, 0, n^2}], {n, 3, 36}]

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^5 / (32 * (1 - x)).

A341400 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n.

Original entry on oeis.org

1, 6, 16, 26, 36, 57, 87, 107, 122, 157, 207, 247, 277, 322, 392, 452, 482, 537, 637, 717, 773, 863, 973, 1053, 1113, 1203, 1343, 1473, 1553, 1668, 1858, 1998, 2053, 2173, 2373, 2543, 2673, 2818, 3018, 3218, 3338, 3483, 3753, 3973, 4113, 4344, 4634, 4834, 4944, 5139, 5449

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A038671.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A038671, A055404, A055411, A175360, A224212, A224213, A302862, A341401, A341402.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 5)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 10 2021

nmax = 50; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^5/(32 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^5 / (32 * (1 - x)).

a(n^2) = A055404(n).

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}]

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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A341424 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n^2.

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A341400 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n.

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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.

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Mathematica