A303333 -id:A303333 - cp's OEIS frontend

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

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

A297331 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 4, 2, 0, 1, 12, 4, 0, 0, 1, 24, 6, 0, 0, 0, 1, 40, 24, 24, 4, 0, 0, 1, 60, 90, 96, 12, 8, 0, 0, 1, 84, 252, 240, 24, 24, 0, 0, 0, 1, 112, 574, 544, 200, 144, 8, 0, 2, 0, 1, 144, 1136, 1288, 1020, 560, 96, 48, 4, 0, 0, 1, 180, 2034, 3136, 3444, 1560, 400, 192, 6, 4, 0, 0

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

			Square array begins:
1,  1,  1,   1,    1,    1,  ...
0,  0,  4,  12,   24,   40,  ...
0,  2,  4,   6,   24,   90,  ...
0,  0,  0,  24,   96,  240,  ...
0,  0,  4,  12,   24,  200,  ...
0,  0,  8,  24,  144,  560,  ...

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 118.

Columns k=0..32 give A000007, A089798 (absolute values), A004018, A004015, A004011, A005930, A008428, A008429, A008430, A008431, A008432, A022042, A022043, A022044, A022045, A022046, A022047, A022048, A022049, A022050, A022051, A022052, A022053, A022054, A022055, A022056, A022057, A022058, A022059, A022060, A022061, A022062, A022063.
Main diagonal gives A303333.
Cf. A122141, A286815.

Table[Function[k, SeriesCoefficient[(EllipticTheta[3, 0, q^(1/2)]^k + EllipticTheta[4, 0, q^(1/2)]^k)/2, {q, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2, where theta_() is the Jacobi theta function.

cp's OEIS Frontend

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

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