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A136028 Expansion of (phi(q) * phi(q^2))^3 in powers of q where phi() is a Ramanujan theta function.

%I A136028 #15 Feb 16 2025 08:33:07
%S A136028 1,6,18,44,90,144,212,288,330,418,528,588,836,1008,1056,1440,1386,
%T A136028 1356,1894,1644,2064,2880,2484,3168,3428,2838,3696,3864,4128,5040,
%U A136028 5280,5760,5418,5656,5988,5376,7678,8208,7572,10080,8208,7788,10560,8652,10404,13104
%N A136028 Expansion of (phi(q) * phi(q^2))^3 in powers of q where phi() is a Ramanujan theta function.
%C A136028 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A136028 Vaclav Kotesovec, <a href="/A136028/b136028.txt">Table of n, a(n) for n = 0..1000</a>
%H A136028 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A136028 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A136028 Expansion of (eta(q^2) * eta(q^4))^9 / (eta(q) * eta(q^8))^6 in powers of q.
%F A136028 G.f. is a period 1 Fourier series which satisfies f(-1/(8 t)) = 2^(9/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
%F A136028 G.f.: (Product_{k>0} (1 - x^k)^2 * (1 + x^k)^4 * (1 + x^(2*k)) / (1 + x^(4*k))^2)^3.
%F A136028 a(n) = A029713(n) + 6 * A030207(n). Convolution of A033715 and A097057.
%F A136028 a(n) = A028578(4*n). - _Michael Somos_, Oct 14 2015
%e A136028 G.f. = 1 + 6*q + 18*q^2 + 44*q^3 + 90*q^4 + 144*q^5 + 212*q^6 + 288*q^7 + ...
%t A136028 nmax=60; CoefficientList[Series[Product[((1-x^k)^2 * (1+x^k)^4 * (1+x^(2*k)) / (1+x^(4*k))^2)^3,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%t A136028 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2])^3, {q, 0, n}]; (* _Michael Somos_, Oct 14 2015 *)
%o A136028 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A) * eta(x^4 + A))^3 / (eta(x + A) * eta(x^8 + A))^2)^3, n))};
%o A136028 (Magma) A := Basis( ModularForms( Gamma1(8), 3), 46);  A[1] + 6*A[2] + 18*A[3] + 44*A[4] + 90*A[5] + 144*A[6] + 212*A[7]; /* _Michael Somos_, Oct 14 2015 */
%Y A136028 Cf. A029713, A030207, A033715, A097057.
%K A136028 nonn
%O A136028 0,2
%A A136028 _Michael Somos_, Dec 10 2007