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A320049 Expansion of (psi(x) / phi(x))^6 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A320049 #16 Feb 16 2025 08:33:56
%S A320049 1,-6,27,-98,309,-882,2330,-5784,13644,-30826,67107,-141444,289746,
%T A320049 -578646,1129527,-2159774,4052721,-7474806,13569463,-24274716,
%U A320049 42838245,-74644794,128533884,-218881098,368859591,-615513678,1017596115,-1667593666,2710062756,-4369417452
%N A320049 Expansion of (psi(x) / phi(x))^6 in powers of x where phi(), psi() are Ramanujan theta functions.
%H A320049 Seiichi Manyama, <a href="/A320049/b320049.txt">Table of n, a(n) for n = 0..10000</a>
%H A320049 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A320049 Convolution inverse of A029843.
%F A320049 Expansion of q^(-3/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^6 in powers of q.
%F A320049 a(n) ~ (-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n)) / (128*sqrt(2)*n^(3/4)). - _Vaclav Kotesovec_, Oct 06 2018
%t A320049 nmax = 40; CoefficientList[Series[Product[((1-x^k) * (1-x^(4*k))^2 / (1-x^(2*k))^3)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 06 2018 *)
%Y A320049 (psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), this sequence (b=6), A320050 (b=7).
%Y A320049 Cf. A029843.
%K A320049 sign
%O A320049 0,2
%A A320049 _Seiichi Manyama_, Oct 04 2018