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A226235 Expansion of q * (chi(-q) / chi(-q^3))^12 in powers of q where chi() is a Ramanujan theta function.

%I A226235 #36 Feb 16 2025 08:33:19
%S A226235 1,-12,66,-220,495,-804,1068,-1596,3279,-6952,12276,-17844,23653,
%T A226235 -34080,57168,-98428,154332,-215724,285388,-395784,600459,-931888,
%U A226235 1365696,-1853076,2426189,-3277896,4689534,-6815008,9538632,-12664440,16403188,-21690876,29812932
%N A226235 Expansion of q * (chi(-q) / chi(-q^3))^12 in powers of q where chi() is a Ramanujan theta function.
%C A226235 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A226235 Seiichi Manyama, <a href="/A226235/b226235.txt">Table of n, a(n) for n = 1..10000</a>
%H A226235 Richard Moy, <a href="http://arxiv.org/abs/1309.4320">Congruences among power series coefficients of modular forms</a>, arXiv:1309.4320 [math.NT], 2013.
%H A226235 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A226235 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A226235 Wenzhe Yang, <a href="https://arxiv.org/abs/1911.02608">Apéry's irrationality proof, mirror symmetry and Beukers' modular forms</a>, arXiv:1911.02608 [math.NT], 2019.
%F A226235 Expansion of q * (f(-q, -q^5) / f(-q^6))^12 in powers of q where f() is a Ramanujan theta function.
%F A226235 Expansion of ((c(q^2) * b(q)) / (c(q) * b(q^2)))^3 in powers of q where b() and c() are cubic AGM theta functions.
%F A226235 Expansion of (eta(q) * eta(q^6) / (eta(q^2) * eta(q^3)))^12 in powers of q.
%F A226235 Euler transform of period 6 sequence [ -12, 0, 0, 0, -12, 0, ...].
%F A226235 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w)= (v - u^2) * (v - w^2) - u*w * (24*(1 + v^2) + 152*v).
%F A226235 G.f. A(x) satisfies f(x) = g(A(x)) where f, g are the g.f. for A006353 and A005259.
%F A226235 G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = f(t) where q = exp(2 Pi i t).
%F A226235 G.f.: x * (Product_{k>0} 1 - x^k + x^(2*k))^12 where 1 - x + x^2 is the 6th cyclotomic polynomial.
%F A226235 Convolution inverse of A121665. Convolution 12th power of A109389.
%F A226235 a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n/3)) / (2*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Mar 30 2017
%F A226235 Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 11 + 5*sqrt(3) - sqrt(189 + 114*sqrt(3)). - _Simon Plouffe_, Mar 02 2021
%e A226235 G.f. = q - 12*q^2 + 66*q^3 - 220*q^4 + 495*q^5 - 804*q^6 + 1068*q^7 - 1596*q^8 + ...
%t A226235 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^6] / (QPochhammer[ q^2] QPochhammer[ q^3]))^12, {q, 0, n}]
%t A226235 nmax = 50; CoefficientList[Series[Product[((1 + x^(3*k))/(1 + x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 30 2017 *)
%o A226235 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A)))^12, n))}
%Y A226235 Cf. A005259, A006353, A109389, A121665.
%K A226235 sign
%O A226235 1,2
%A A226235 _Michael Somos_, Sep 18 2013