A287617 -id:A287617 - cp's OEIS frontend

A298330 Number of ordered ways of writing n^2 as a sum of n squares of positive integers.

1, 1, 0, 3, 1, 5, 141, 742, 6120, 43888, 300232, 3074478, 28901797, 290411147, 3175037698, 34951274416, 399750066121, 4814421349467, 59532792202344, 768079420764884, 10247011240209066, 140144002390928732, 1978092111496441512, 28633995987157024399

Offset: 0

Ilya Gutkovskiy, Jan 17 2018

nonn

			a(3) = 3 because we have [4, 4, 1], [4, 1, 4] and [1, 4, 4].

Robert Israel, Table of n, a(n) for n = 0..100
Index entries for sequences related to sums of squares

Cf. A037444, A066535, A232173, A287617, A298329.

G:= (JacobiTheta3(0,x)-1)/2:
f:= proc(n) local S; S:= series(G^n,x,n^2+1); coeff(S,x,n^2) end proc:
map(f, [$0..25]); # Robert Israel, Dec 16 2024

Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^2}], {n, 0, 23}]

a(n) = [x^(n^2)] (Sum_{k>=1} x^(k^2))^n.

A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 1, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 2, 0, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 6, 2, 0, 0

Offset: 0

N. J. A. Sloane

			Rows are
1,0,0,..;
1,1,0,0,1,0..;
1,2,1,0,2,2,..;
1,3,3,1,...

Seiichi Manyama, Ascending antidiagonals n = 0..139, flattened
H. Wilf, A combinatorial determinant, arXiv:math/9809120 [math.CO], 1998.

Rows k=0-10 give: A000007, A010052, A000925, A002102, A014110, A038671, A045848, A045849, A045850, A045851, A045852.
Diagonal gives A287617.
Antidiagonal sums give A302018.

i-th row is expansion of (1+x+x^4+x^9+...)^i.

More terms from Erich Friedman

A298329 Number of ordered ways of writing n^2 as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 2, 6, 5, 90, 582, 4081, 45678, 378049, 3844532, 39039539, 395170118, 4589810849, 53154371025, 660113986997, 8584476248237, 113555197832758, 1572878837435750, 22259911738401660, 324143769099772448, 4869443438412466557, 74837370448784241452, 1182177603062005007658

Offset: 0

Ilya Gutkovskiy, Jan 17 2018

nonn

			a(3) = 6 because we have [9, 0, 0], [4, 4, 1], [4, 1, 4], [1, 4, 4], [0, 9, 0] and [0, 0, 9].

Alois P. Heinz, Table of n, a(n) for n = 0..100 (first 81 terms from Seiichi Manyama)
Index entries for sequences related to sums of squares

[x^(n^b)] (Sum_{k>=0} x^(k^b))^n: A088218 (b=1), this sequence (b=2), A298671 (b=3).

Cf. A037444, A066535, A232173, A287617, A298330, A300446.

b:= proc(n, i, t) option remember; `if`(n=0, 1/t!, (s->
     `if`(s*t n!*b(n^2, n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 28 2018

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^2}], {n, 0, 23}]

{a(n) = polcoeff((sum(k=0, n, x^(k^2)+x*O(x^(n^2))))^n, n^2)} \\ Seiichi Manyama, Oct 28 2018

a(n) = [x^(n^2)] (Sum_{k>=0} x^(k^2))^n.

A291700 Number of ways of writing n as a sum of n nonnegative cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 9, 73, 361, 1321, 3961, 10297, 24025, 51481, 103081, 196521, 368425, 720937, 1589161, 4069801, 11511721, 33341353, 94142313, 253860201, 650564201, 1588228228, 3716917597, 8418378043, 18699454621, 41451042556, 93508305513, 218218347865, 530189399785

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for sequences related to sums of cubes

Main diagonal of A290054.

Cf. A066535, A106337, A287617.

Table[SeriesCoefficient[Sum[x^k^3, {k, 0, n}]^n, {x, 0, n}], {n, 0, 34}]

a(n) = [x^n] (Sum_{k>=0} x^(k^3))^n.

A302862 a(n) = [x^n] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 4, 8, 20, 57, 160, 422, 1076, 2780, 7449, 20462, 56348, 153909, 418268, 1139703, 3126068, 8618611, 23801146, 65708424, 181391905, 501296216, 1387834518, 3848187985, 10680579812, 29660831057, 82415406493, 229156296047, 637659848888, 1775648562970, 4947475298595

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.

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

Cf. A000606, A003059, A010052, A224212, A224213, A287617, A302860, A302863.

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/(2^n (1 - x)), {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 0, n}]^n, {x, 0, n}], {n, 0, 30}]

A298938 Number of ordered ways of writing n^3 as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 4, 5, 686, 13942, 455988, 13617853, 454222894, 18323165948, 802161109047, 42149084452070, 2481730049781672, 157265294178424356, 10977302934685469078, 812821237985857557677, 64539935903231450294134, 5504599828399250884049308

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(4) = 5 because we have [64, 0, 0, 0], [16, 16, 16, 16], [0, 64, 0, 0], [0, 0, 64, 0] and [0, 0, 0, 64].

Index entries for sequences related to sums of squares

Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298936, A298939.

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}]

a(n) = [x^(n^3)] (Sum_{k>=0} x^(k^2))^n.

A298939 Number of ordered ways of writing n^3 as a sum of n squares of positive integers.

Original entry on oeis.org

1, 1, 1, 4, 1, 286, 7582, 202028, 6473625, 226029577, 8338249868, 391526193477, 19990594900630, 1159906506684446, 74890158861242740, 5119732406649036418, 380146984328280974281, 30198665638519565614034, 2555354508318427693497565

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(3) = 4 because we have [25, 1, 1], [9, 9, 9], [1, 25, 1] and [1, 1, 25].

Index entries for sequences related to sums of squares

Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298937, A298938.

Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}]

a(n) = [x^(n^3)] (Sum_{k>=1} x^(k^2))^n.

A338464 Number of ways to write 2*n as an ordered sum of n squares of positive integers.

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 15, 0, 8, 84, 0, 110, 495, 0, 1092, 3018, 120, 9520, 18870, 2907, 77520, 120270, 43890, 606188, 780023, 531300, 4620200, 5161377, 5651100, 34622172, 35045340, 55234560, 256503672, 245772464, 508930224, 1886151225, 1788167610, 4491607230

Offset: 0

Ilya Gutkovskiy, Jan 31 2021

nonn

Also number of ways to write n as an ordered sum of n nonnegative numbers one less than a square.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000290, A005563, A066535, A111178, A287617, A303333, A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..39);  # Alois P. Heinz, Feb 04 2021

Table[SeriesCoefficient[((EllipticTheta[3, 0, x] - 1)/2)^n, {x, 0, 2 n}], {n, 0, 37}]

a(n) = [x^(2*n)] ((theta_3(x) - 1) / 2)^n, where theta_3() is the Jacobi theta function.
a(n) = [x^n] (Sum_{k>=0} x^(k*(k + 2)))^n.
a(n) = A337165(2n,n). - Alois P. Heinz, Feb 04 2021

A303172 Number of ordered ways of writing n as a sum of n square pyramidal numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 31, 106, 281, 631, 1306, 2806, 6931, 19306, 55070, 150816, 391161, 977501, 2426071, 6141865, 16000186, 42465571, 112950916, 297793651, 776866355, 2015237231, 5233754306, 13668689206, 35908153534, 94633042267, 249398115466, 656105299636, 1723150461561

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Main diagonal of A290430.

Cf. A000330, A066535, A279220, A287617, A303170.

Table[SeriesCoefficient[Sum[x^(k (k + 1) (2 k + 1)/6), {k, 0, n}]^n, {x, 0, n}], {n, 0, 32}]

a(n) = [x^n] (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^n.

a(n) = A290430(n,n).

A363780 a(n) = [x^n] 1/(Sum_{k>=0} x^(k^2))^n.

Original entry on oeis.org

1, -1, 3, -10, 31, -96, 294, -876, 2511, -6796, 16698, -33540, 31174, 184534, -1627812, 8912760, -41466433, 176963760, -714194382, 2766892840, -10374065814, 37815483948, -134334781732, 465432203640, -1571910265770, 5164302815179, -16438631981418

Offset: 0

Seiichi Manyama, Jun 21 2023

sign

Main diagonal of A363778.

Cf. A287617.

cp's OEIS Frontend

A298330 Number of ordered ways of writing n^2 as a sum of n squares of positive integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A298329 Number of ordered ways of writing n^2 as a sum of n squares of nonnegative integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A291700 Number of ways of writing n as a sum of n nonnegative cubes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A302862 a(n) = [x^n] (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

A298938 Number of ordered ways of writing n^3 as a sum of n squares of nonnegative integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A298939 Number of ordered ways of writing n^3 as a sum of n squares of positive integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A338464 Number of ways to write 2*n as an ordered sum of n squares of positive integers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple