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

%I A260671 #23 Jul 31 2018 00:29:20
%S A260671 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,
%T A260671 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,
%U A260671 0,4,0,0,0,0,0,0,4,0,0,4,0,2,0,0,0,4,0
%N A260671 Expansion of theta_3(q) * theta_3(q^15) in powers of q.
%C A260671 a(n) is the number of solutions in integers (x, y) of x^2 + 15*y^2 = n. - _Michael Somos_, Jul 17 2018
%H A260671 G. C. Greubel, <a href="/A260671/b260671.txt">Table of n, a(n) for n = 0..2500</a>
%F A260671 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.
%F A260671 Euler transform of a period 60 sequence.
%F A260671 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).
%F A260671 G.f.: (Sum_{k in Z} x^(k^2)) * (Sum_{k in Z} x^(15*k^2)).
%F A260671 a(3*n + 2) = a(4*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.
%F A260671 a(4*n) = A028625(n). a(4*n + 1) = 2 * A260675(n). a(4*n + 3) = 2 * A260676(n).
%F A260671 a(5*n) = A192323(n).
%F A260671 a(n) = A122855(n) + A140727(n).
%e A260671 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 + ...
%t A260671 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^15], {q, 0, n}];
%o A260671 (PARI) {a(n) = if( n<1, n==0, qfrep([1, 0; 0, 15], n)[n]*2)};
%o A260671 (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))};
%o A260671 (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
%Y A260671 Cf. A028625, A122855, A140727, A192323, A260675, A260676.
%K A260671 nonn
%O A260671 0,2
%A A260671 _Michael Somos_, Nov 14 2015