A055409 -id:A055409 - 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.

A341429 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n^2.

Original entry on oeis.org

56, 1108, 12098, 84624, 439371, 1785368, 6078048, 18139393, 48586117, 118929400, 270250734, 578320470, 1169522013, 2261784392, 4193751331, 7509793133, 13008356489, 21921125415, 35951569269, 57666975238, 90464266824, 139295784464, 210514511189, 313228848537

Offset: 4

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000
Index entries for sequences related to sums of squares

Cf. A000122, A001182, A055409, A055416, A253663, A302995, A340947, A341399, A341405, A341423, A341424, A341425, A341426, A341427, A341428.

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, 10):
seq(a(n), n=4..27);  # Alois P. Heinz, Feb 11 2021

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^10/(1024 (1 - x)), {x, 0, n^2}], {n, 4, 27}]

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^10 / (1024 * (1 - x)).

A341399 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

Original entry on oeis.org

1, 21, 201, 1161, 4541, 12965, 29285, 58085, 110105, 198765, 327829, 503509, 765589, 1152509, 1642109, 2243069, 3083569, 4221529, 5551949, 7115789, 9166133, 11777333, 14763893, 18121973, 22316213, 27634481, 33512921, 39812441, 47674841, 57294401, 67510721, 78592961

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A000144.

Index entries for sequences related to sums of squares

Cf. A000122, A000144, A001650, A046895, A055409, A055416, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341397, A341398.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 10)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..31);  # Alois P. Heinz, Feb 10 2021

nmax = 31; CoefficientList[Series[EllipticTheta[3, 0, x]^10/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[10, n], {n, 0, 31}] // Accumulate

G.f.: theta_3(x)^10 / (1 - x).

a(n^2) = A055416(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}]

A341405 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

Original entry on oeis.org

1, 11, 56, 176, 396, 738, 1308, 2268, 3618, 5258, 7449, 10689, 14889, 19609, 25369, 33289, 43154, 53774, 65739, 81339, 100671, 121221, 143421, 171501, 205701, 241283, 278678, 324398, 378998, 435968, 495428, 566468, 650798, 737888, 826083, 930123, 1053323

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045852.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045852, A055409, A055416, A224212, A224213, A302862, A341400, A341401, A341402, A341403, A341404.

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, 10)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 10 2021

nmax = 36; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^10/(1024 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^10 / (1024 * (1 - x)).

a(n^2) = A055409(n).

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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A341429 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n^2.

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A341399 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

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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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Programs

Mathematica

A341405 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula