A260671 Expansion of theta_3(q) * theta_3(q^15) in powers of q.

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 6, 0, 0, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 10, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0

Offset

0,2

Comments

a(n) is the number of solutions in integers (x, y) of x^2 + 15*y^2 = n. - Michael Somos, Jul 17 2018

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Formula

Expansion of (eta(q^2) * eta(q^30))^5 / (eta(q) * eta(q^4) * eta(q^15) * eta(q^60))^2 in powers of q.

Euler transform of a period 60 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = 60^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

G.f.: (Sum_{k in Z} x^(k^2)) * (Sum_{k in Z} x^(15*k^2)).

a(3*n + 2) = a(4*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.

a(4*n) = A028625(n). a(4*n + 1) = 2 * A260675(n). a(4*n + 3) = 2 * A260676(n).

a(5*n) = A192323(n).

a(n) = A122855(n) + A140727(n).

Example

G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^15 + 6*x^16 + 4*x^19 + 4*x^24 + 2*x^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^15], {q, 0, n}];

Prog

(PARI) {a(n) = if( n<1, n==0, qfrep([1, 0; 0, 15], n)[n]*2)};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^30 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^15 + A) * eta(x^60 + A))^2, n))};
(PARI) q='q+O('q^99); Vec((eta(q^2)*eta(q^30))^5/(eta(q)*eta(q^4)*eta(q^15)*eta(q^60))^2) \\ Altug Alkan, Jul 18 2018

Crossrefs

Cf. A028625, A122855, A140727, A192323, A260675, A260676.

Sequence in context: A227395 A255258 A329266 * A033725 A204010 A033723

Adjacent sequences: A260668 A260669 A260670 * A260672 A260673 A260674

Keyword

nonn

Author

Michael Somos, Nov 14 2015

Status

approved