A028625 Expansion of (theta_3(z)*theta_3(15z)+theta_2(z)*theta_2(15z)).

1, 2, 0, 0, 6, 0, 4, 0, 0, 2, 4, 0, 0, 0, 0, 2, 10, 0, 0, 4, 0, 0, 0, 0, 8, 2, 0, 0, 0, 0, 0, 4, 0, 0, 8, 0, 6, 0, 0, 0, 8, 0, 0, 0, 0, 0, 8, 0, 0, 2, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 6, 4, 0, 0, 14, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 12, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 12, 0, 0, 0, 6, 0, 0, 0

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Theta series of quadratic form (or lattice) with Gram matrix [ 2, 1; 1, 8 ].

The number of integer solutions (x, y) to x^2 + x*y + 4*y^2 = n, discriminant -15. - Ray Chandler, Jul 12 2014

a(n) = number of solutions in integers (x, y) of x^2 + 15*y^2 = 4*n. - Michael Somos, Jul 17 2018

Links

John Cannon, Table of n, a(n) for n = 0..5000

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of phi(q) * phi(q^15) + 4 * q^4 * psi(q^2) * psi(q^30) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 26 2006

Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q)* eta(q^15)) + (eta(q) *eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q. - Michael Somos, Aug 26 2006

a(n) = A260671(4*n). - Michael Somos, Jul 17 2018

Example

G.f. = 1 + 2*q^2 + 6*q^8 + 4*q^12 + 2*q^18 + 4*q^20 + 2*q^30 + 10*q^32 + 4*q^38 + 8*q^48 + 2*q^50 + 4*q^62 + 8*q^68 + 6*q^72 + 8*q^80 + 8*q^92 + 2*q^98 + ...
G.f. = 1 + 2*x + 6*x^4 + 4*x^6 + 2*x^9 + 4*x^10 + 2*x^15 + 10*x^16 + 4*x^19 + ... - Michael Somos, Jul 17 2018

Mathematica

r[n_] := Reduce[x^2 + x*y + 4*y^2 == n, {x, y}, Integers]; Table[rn = r[n]; Which[rn === False, 0, Head[rn] === Or, Length[rn], Head[rn] === And, 1], {n, 0, 105}] (* Jean-François Alcover, Nov 05 2015, after the comment by Ray Chandler *)
a[0] = 1; a[n_] := With[{K = KroneckerSymbol}, DivisorSum[n, K[-15, #] + K[#, 3]*K[n/#, 5]&]]; Table[a[n], {n, 0, 103}] (* Jean-François Alcover, Jul 07 2017, after Michael Somos *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^15] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^15], {x, 0, n}]; (* Michael Somos, Jul 17 2018 *)

Prog

(PARI) {a(n) = if( n<1, n==0, qfrep([2, 1; 1, 8], n, 1)[n]*2)}; /* Michael Somos, Aug 26 2006 */
(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-15, d) + kronecker(d, 3) * kronecker(n/d, 5) ))}; /* Michael Somos, Aug 26 2006 */

Crossrefs

Cf. A260671.

Sequence in context: A241020 A277444 A274710 * A344441 A221728 A339942

Adjacent sequences: A028622 A028623 A028624 * A028626 A028627 A028628

Keyword

nonn

Author

N. J. A. Sloane

Status

approved