A055412 -id:A055412 - 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.

A175361 Partial sums of A000141.

Original entry on oeis.org

1, 13, 73, 233, 485, 797, 1341, 2301, 3321, 4197, 5757, 8157, 10237, 12277, 15541, 19701, 23793, 27273, 31653, 38853, 45405, 50013, 58173, 68733, 76957, 84769, 94969, 108089, 120569, 130673, 144817, 164017, 180397, 191917, 209317, 234277, 252673, 269113, 293593

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

The 6th row of A122510.

Cf. A000141, A055412, A122510.

CoefficientList[Series[EllipticTheta[3,x]^6/(1-x),{x,0,35}],x] (* Stefano Spezia, Jun 21 2024 *)

from math import prod
from sympy import factorint
def A175361(n):
    c = 1
    for m in range(1,n+1):
        f = [(p,e,(0,1,0,-1)[p&3]) for p,e in factorint(m).items()]
        c += (prod((p**(e+1<<1)-a)//(p**2-a) for p, e, a in f)<<2)-prod(((k:=p**2*a)**(e+1)-1)//(k-1) for p, e, a in f)<<2
    return c # Chai Wah Wu, Jun 21 2024

a(n^2) = A055412(n).

G.f.: theta_3(x)^6 / (1 - x). - Ilya Gutkovskiy, Feb 13 2021

A341425 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n^2.

Original entry on oeis.org

7, 48, 331, 1269, 3698, 9382, 20927, 42683, 79844, 142173, 238810, 387615, 603589, 915324, 1345294, 1939221, 2729723, 3783313, 5138567, 6895632, 9108626, 11909496, 15362753, 19642539, 24832744, 31179476, 38757032, 47877886, 58647957, 71447776, 86391220

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, A055405, A055412, A175361, A253663, A302995, A340905, A341401, A341423, A341424, 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, 6):
seq(a(n), n=3..33);  # Alois P. Heinz, Feb 11 2021

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^6/(64 (1 - x)), {x, 0, n^2}], {n, 3, 33}]

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^6 / (64 * (1 - x)).

A341401 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n.

Original entry on oeis.org

1, 7, 22, 42, 63, 99, 160, 220, 265, 337, 457, 577, 672, 792, 978, 1178, 1319, 1469, 1739, 2039, 2255, 2507, 2882, 3242, 3513, 3819, 4269, 4769, 5159, 5555, 6181, 6841, 7246, 7666, 8401, 9181, 9763, 10363, 11188, 12108, 12828, 13434, 14394, 15534, 16359, 17211, 18477, 19677

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045848.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045848, A055405, A055412, A175361, A224212, A224213, A302862, A341400, 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, 6)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..47);  # Alois P. Heinz, Feb 10 2021

nmax = 47; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^6/(64 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^6 / (64 * (1 - x)).

a(n^2) = A055405(n).

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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A175361 Partial sums of A000141.

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A341425 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n^2.

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A341401 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= 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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