cp's OEIS Frontend

A028613 Expansion of theta_3(q) * theta_3(q^12) + theta_2(q) * theta_2(q^12) in powers of q^(1/4).

%I A028613 #14 Jul 08 2025 18:40:24
%S A028613 1,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,2,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,
%T A028613 0,0,2,4,0,0,0,0,0,0,0,0,0,0,2,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,6,0,0,0,
%U A028613 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0
%N A028613 Expansion of theta_3(q) * theta_3(q^12) + theta_2(q) * theta_2(q^12) in powers of q^(1/4).
%F A028613 a(4*n + 2) = a(4*n + 3) = a(8*n + 1) = a(16*n + 8) = a(16*n + 12) = 0. - _Michael Somos_, Feb 22 2015
%F A028613 a(8*n + 5) = 4*A112607(n-1). a(16*n) = A033716(n). a(16*n + 4) = 2*A112604(n). - _Michael Somos_, Feb 22 2015
%e A028613 G.f. = 1 + 2*x^4 + 4*x^13 + 2*x^16 + 4*x^21 + 2*x^36 + 4*x^37 + 2*x^48 + ...
%e A028613 G.f. = 1 + 2*q + 4*q^(13/4) + 2*q^4 + 4*q^(21/4) + 2*q^9 + 4*q^(37/4) + 2*q^12 + ...
%t A028613 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^12] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^12], {x, 0, n/4}]; (* _Michael Somos_, Feb 22 2015 *)
%o A028613 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) * eta(x^96+A))^5 / (eta(x^4 + A) * eta(x^16 + A) * eta(x^48 + A) * eta(x^192 + A))^2 + 4*x^13 * (eta(x^16 + A) * eta(x^192 + A))^2 / (eta(x^8 + A) * eta(x^96 + A)), n))};
%Y A028613 Cf. A033716, A112604, A112607.
%K A028613 nonn
%O A028613 0,5
%A A028613 _N. J. A. Sloane_