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A328798 Expansion of 1 / (chi(-x) * chi(-x^3)) in powers of x where chi() is a Ramanujan theta function.

%I A328798 #17 Feb 16 2025 08:33:58
%S A328798 1,1,1,3,3,4,7,8,10,16,19,23,33,39,48,65,77,93,122,144,173,220,259,
%T A328798 309,384,451,534,653,764,899,1085,1264,1479,1765,2048,2385,2820,3260,
%U A328798 3778,4432,5105,5891,6864,7879,9056,10491,12002,13744,15839,18064,20616,23648
%N A328798 Expansion of 1 / (chi(-x) * chi(-x^3)) in powers of x where chi() is a Ramanujan theta function.
%C A328798 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A328798 Convolution inverse is A112175, 2nd power is A102315, 3rd power is A229180, 6th power is A123653.
%C A328798 f(-1 / (216 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A112175.
%H A328798 Vaclav Kotesovec, <a href="/A328798/b328798.txt">Table of n, a(n) for n = 0..10000</a>
%H A328798 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A328798 Expansion of q^(-1/6) * eta(q^2) * eta(q^6) / (eta(q) * eta(q^3)) in powers of q.
%F A328798 Euler transform of period 6 sequence [1, 0, 2, 0, 1, 0, ...].
%F A328798 G.f.: Product_{k>=1} (1 + x^k)^(-1) * (1 + x^(3*k))^(-1).
%F A328798 a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Oct 31 2019
%e A328798 G.f. = 1 + x + x^2 + 3*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 8*x^7 + ...
%e A328798 G.f. = q + q^7 + q^13 + 3*q^19 + 3*q^25 + 4*q^31 + 7*q^37 + ...
%t A328798 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^3, x^3], {x, 0, n}];
%o A328798 (PARI) {a(n) = my(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) / (eta(x + A) * eta(x^3 + A)), n))};
%Y A328798 Cf. A102315, A112175, A123653, A229180.
%K A328798 nonn
%O A328798 0,4
%A A328798 _Michael Somos_, Oct 28 2019