A033723 -id:A033723 - cp's OEIS frontend

A317641 Expansion of theta_3(q)*theta_3(q^10), where theta_3() is the Jacobi theta function.

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 4, 0, 0, 4, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 2, 4, 0, 0, 4, 0, 4, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 2, 8, 0, 0, 4, 0, 0, 0, 0, 4, 2

Offset: 0

Ilya Gutkovskiy, Aug 02 2018

nonn

Number of integer solutions to the equation x^2 + 10*y^2 = n.

			G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^10 + 4*q^11 + 4*q^14 + 2*q^16 + 4*q^19 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A000024, A020673, A033201, A033723, A216577, A216579, A258034.

nmax = 100; CoefficientList[Series[EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^10], {q, 0, nmax}], q]
nmax = 100; CoefficientList[Series[QPochhammer[-q, -q] QPochhammer[-q^10, -q^10]/(QPochhammer[q, -q] QPochhammer[q^10, -q^10]), {q, 0, nmax}], q]

G.f.: Product_{k>=1} (1 + x^(2*k-1))^2*(1 - x^(2*k))*(1 + x^(20*k-10))^2*(1 - x^(20*k)).

A306518 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{d|k} theta_3(q^d).

Original entry on oeis.org

1, 1, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 0, 4, 2, 1, 2, 2, 2, 2, 0, 1, 2, 0, 4, 6, 0, 0, 1, 2, 2, 0, 4, 0, 4, 0, 1, 2, 0, 6, 2, 4, 0, 0, 0, 1, 2, 2, 0, 6, 2, 8, 4, 2, 2, 1, 2, 0, 4, 2, 4, 4, 8, 0, 6, 0, 1, 2, 2, 2, 4, 0, 14, 0, 6, 2, 0, 0, 1, 2, 0, 4, 6, 4, 0, 8, 0, 6, 0, 4, 0, 1, 2, 2, 0, 2, 0, 8, 2, 6, 6, 8, 0, 4, 0

Offset: 0

Ilya Gutkovskiy, Feb 21 2019

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  2,  2,  2,  2,  2,  2,  ...
  0,  2,  0,  2,  0,  2,  ...
  0,  4,  2,  4,  0,  6,  ...
  2,  2,  6,  4,  2,  6,  ...
  0,  0,  0,  4,  2,  4,  ...

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Columns k=1..48 give A000122, A033715, A033716, A033717, A033718, A033712, A033719, A033720, A033721, A033722, A033723, A033724, A033725, A033726, A033727, A033728, A033729, A033730, A033731, A033732, A033733, A033734, A033735, A033736, A033737, A033738, A033739, A033740, A033741, A033742, A033743, A033744, A033745, A033746, A033747, A033748, A033749, A033750, A033751, A033752, A033753, A033754, A033755, A033756, A033757, A033758, A033759, A033760.

Cf. A320305 (diagonal).

Table[Function[k, SeriesCoefficient[Product[EllipticTheta[3, 0, q^d], {d, Divisors[k]}], {q, 0, n}]][i - n + 1], {i, 0, 13}, {n, 0, i}] // Flatten

G.f. of column k: Product_{d|k} theta_3(q^d).

cp's OEIS Frontend

A317641 Expansion of theta_3(q)*theta_3(q^10), where theta_3() is the Jacobi theta function.

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