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

%I A028625 #26 Feb 16 2025 08:32:35
%S A028625 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,
%T A028625 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,
%U A028625 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
%N A028625 Expansion of (theta_3(z)*theta_3(15z)+theta_2(z)*theta_2(15z)).
%C A028625 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A028625 Theta series of quadratic form (or lattice) with Gram matrix [ 2, 1; 1, 8 ].
%C A028625 The number of integer solutions (x, y) to x^2 + x*y + 4*y^2 = n, discriminant -15. - _Ray Chandler_, Jul 12 2014
%C A028625 a(n) = number of solutions in integers (x, y) of x^2 + 15*y^2 = 4*n. - _Michael Somos_, Jul 17 2018
%H A028625 John Cannon, <a href="/A028625/b028625.txt">Table of n, a(n) for n = 0..5000</a>
%H A028625 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H A028625 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A028625 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A028625 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
%F A028625 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
%F A028625 a(n) = A260671(4*n). - _Michael Somos_, Jul 17 2018
%e A028625 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 + ...
%e A028625 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
%t A028625 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_ *)
%t A028625 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_ *)
%t A028625 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 *)
%o A028625 (PARI) {a(n) = if( n<1, n==0, qfrep([2, 1;1, 8], n, 1)[n]*2)}; /* _Michael Somos_, Aug 26 2006 */
%o A028625 (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 */
%Y A028625 Cf. A260671.
%K A028625 nonn
%O A028625 0,2
%A A028625 _N. J. A. Sloane_