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A235870 Expansion of ( f(-q)^12 + 22 * q * f(-q)^6 * f(-q^5)^6 + 125 * q^2 * f(-q^5)^12 ) / (f(-q) * f(-q^5))^2 in powers of q where f() is a Ramanujan theta function.

%I A235870 #21 Feb 16 2025 08:33:21
%S A235870 1,12,72,264,696,1380,2304,3192,5400,6924,12600,12384,18912,20184,
%T A235870 28512,39000,43032,45432,63144,63600,101640,88944,110304,112104,
%U A235870 151200,174540,183024,188400,231936,225000,351360,274704,346392,344448,407952,479400,509592
%N A235870 Expansion of ( f(-q)^12 + 22 * q * f(-q)^6 * f(-q^5)^6 + 125 * q^2 * f(-q^5)^12 ) / (f(-q) * f(-q^5))^2 in powers of q where f() is a Ramanujan theta function.
%C A235870 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A235870 G. C. Greubel, <a href="/A235870/b235870.txt">Table of n, a(n) for n = 0..1000</a>
%H A235870 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A235870 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A235870 Expansion of ( ( (f(-q) * f(-q^5))^4 + 9*q * (f(-q) * f(-q^3) * f(-q^5) * f(-q^15))^2 + 27*q * (f(-q^3) * f(-q^15))^4 ) / (f(-q) * f(-q^3) * f(-q^5) * f(-q^15)) )^2 in powers of q where f() is a Ramanujan theta function.
%F A235870 G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 25 (t/i)^4 f(t) where q = exp(2 Pi i t).
%F A235870 Convolution square of A028887.
%e A235870 G.f. = 1 + 12*q + 72*q^2 + 264*q^3 + 696*q^4 + 1380*q^5 + 2304*q^6 + ...
%o A235870 (PARI) {a(n) = my(A, u1, u5); if( n<0, 0, A = x * O(x^n); u1 = eta(x + A); u5 = eta(x^5 + A); polcoeff( ( u1^12 + 22*x * (u1 * u5)^6 + 125*x^2 * u5^12 ) / (u1 * u5)^2, n))};
%o A235870 (PARI) {a(n) = my(A, v1, v3); if( n<0, 0, A = x * O(x^n); v1 = eta(x + A) * eta(x^5 + A) ; v3 = eta(x^3 + A) * eta(x^15 + A) ; polcoeff( ( v1^4 + 9*x * (v1 * v3)^2 + 27*x^2 * v3^4 )^2 / (v1 * v3)^2, n))};
%o A235870 (Sage) A = ModularForms( Gamma0(5), 4, prec=36) . basis(); A[1] + 12/13 * (3*A[0] + 10*A[2]); # _Michael Somos_, Jun 13 2014
%o A235870 (Magma) A := Basis( ModularForms( Gamma0(5), 4), 36); A[1] + 12*A[2] + 72*A[3]; /* _Michael Somos_, Jun 13 2014 */
%Y A235870 Cf. A028887.
%K A235870 nonn
%O A235870 0,2
%A A235870 _Michael Somos_, Jun 13 2014