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A028625
Expansion of (theta_3(z)*theta_3(15z)+theta_2(z)*theta_2(15z)).
2
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
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
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
KEYWORD
nonn
STATUS
approved