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A028572 Expansion of theta_3(z)*theta_3(2z) + theta_2(z)*theta_2(2z) in powers of q^(1/4).

%I A028572 #22 Feb 16 2025 08:32:35
%S A028572 1,0,0,4,2,0,0,0,2,0,0,4,4,0,0,0,2,0,0,4,0,0,0,0,4,0,0,8,0,0,0,0,2,0,
%T A028572 0,0,6,0,0,0,0,0,0,4,4,0,0,0,4,0,0,8,0,0,0,0,0,0,0,4,0,0,0,0,2,0,0,4,
%U A028572 4,0,0,0,6,0,0,4,4,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,12,2,0,0,0,0
%N A028572 Expansion of theta_3(z)*theta_3(2z) + theta_2(z)*theta_2(2z) in powers of q^(1/4).
%C A028572 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A028572 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A028572 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A028572 Expansion of phi(x^4) * phi(x^8) + 4 * x^3 * psi(x^8) * psi(x^16) in powers of x where phi(), psi() are Ramanujan theta functions.
%F A028572 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) (t/i) f(t) where q = exp(2 pi i t). - _Michael Somos_, Mar 23 2012
%F A028572 G.f.: Sum_{n,m} x^(3*(n^2 + m^2) + 2*n*m). - _Michael Somos_, Nov 20 2006
%F A028572 a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = 0. a(4*n) = A033715(n). a(8*n + 3) = 4 * A033761(n). - _Michael Somos_, Mar 23 2012
%e A028572 1 + 4*x^3 + 2*x^4 + 2*x^8 + 4*x^11 + 4*x^12 + 2*x^16 + 4*x^19 + 4*x^24 + ...
%e A028572 1 + 4*q^(3/4) +2*q +2*q^2 +4*q^(11/4) +4*q^3 +2*q^4 + 4*q^(19/4) +4*q^6 + ...
%t A028572 terms = 105; max = Sqrt[terms] // Ceiling; s = Sum[x^(3*(n^2 + m^2) + 2*n*m), {n, -max, max}, {m, -max, max}]; CoefficientList[s, x][[1 ;; terms]] (* _Jean-François Alcover_, Dec 03 2015, using 2nd g.f. *)
%o A028572 (PARI) {a(n) = if( n<1, n==0, qfrep( [3, 1; 1, 3], n)[n] * 2)} /* _Michael Somos_, Nov 20 2006 */
%o A028572 (PARI) {a(n) = if( n<1, n==0, if( n%4==1 || n%4==2, 0, 2 * sumdiv( n, d, kronecker( -2, d))))} /* _Michael Somos_, Mar 23 2012 */
%Y A028572 Cf. A033715, A033761.
%K A028572 nonn
%O A028572 0,4
%A A028572 _N. J. A. Sloane_