A055411 -id:A055411 - cp's OEIS frontend

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.

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

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

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

			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,  ...

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, A005408, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416.
Rows k=0..10 give A000012, A005408, A055426, A055427, A055428, A055429, A055430, A055431, A055432, A055433, A055434.
Main diagonal gives A302861.
Cf. A000122, A122510, A302996, A302998.

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

T(n,k)={if(k==0, 1, polcoef(((1 + 2*sum(j=1, 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=-infinity..infinity} x^(j^2))^k.

A175360 Partial sums of A000132.

Original entry on oeis.org

1, 11, 51, 131, 221, 333, 573, 893, 1093, 1343, 1903, 2463, 2863, 3423, 4223, 5183, 5913, 6393, 7633, 9153, 9905, 11025, 12865, 14465, 15665, 16875, 18875, 21115, 22715, 24395, 27115, 30315, 31795, 33235, 36915, 39955, 42205, 45005, 48285

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

The 5th row of A122510.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A000132, A055411, A122510.

# uses Python code for A046895
from math import isqrt
def A175360(n): return A046895(n)+(sum(A046895(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 23 2024

a(n^2) = A055411(n).

G.f.: theta_3(x)^5 / (1 - x). - Ilya Gutkovskiy, Feb 13 2021

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).

A373883 Number of lattice points inside or on the 5-dimensional hypersphere x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 10^n.

Original entry on oeis.org

11, 1903, 532509, 166711479, 52646439609, 16645828150193, 5263797438037625, 1664556518763850069, 526378909839312477785, 166455624316184206850205, 52637890147973140623040513, 16645562406807092052281075983, 5263789013922669372094091725857

Offset: 0

Seiichi Manyama, Jun 21 2024

Cf. A068785, A373881, A373882, A373884, A373885, A373896.

Cf. A055411, A175360.

b(k, n) = my(q='q+O('q^(n+1))); polcoef((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^k/(1-q), n);
a(n) = b(5, 10^n);

a(n) = A175360(10^n).

a(7) from Chai Wah Wu, Jun 22 2024
a(8)-a(10) from Chai Wah Wu, Jun 23 2024
a(11)-a(12) from Chai Wah Wu, Jun 24 2024

A302861 a(n) = [x^(n^2)] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 3, 13, 123, 1281, 16875, 252673, 4031123, 70554353, 1318315075, 26107328109, 549772933959, 12147113355505, 280978137279483, 6780378828922333, 169829490474843659, 4409771551548703649, 118361723203178140163, 3277041835527134201777, 93455465161026267454527

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius n.

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

Main diagonal of A302997.

Cf. A000122, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416, A122510, A232173, A302860.

Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n^2}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n^2}], {n, 0, 19}]

a(n) = A122510(n,n^2).

cp's OEIS Frontend

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.

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A175360 Partial sums of A000132.

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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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A373883 Number of lattice points inside or on the 5-dimensional hypersphere x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 10^n.

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A302861 a(n) = [x^(n^2)] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

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