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A258099 Expansion of ( psi(x^3) * phi(-x^3) / (psi(x) * f(-x^2)) )^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

%I A258099 #16 Feb 16 2025 08:33:25
%S A258099 1,-2,5,-12,26,-50,92,-168,295,-496,818,-1332,2126,-3324,5126,-7824,
%T A258099 11793,-17548,25857,-37788,54734,-78578,111968,-158496,222842,-311224,
%U A258099 432095,-596676,819504,-1119624,1522282,-2060448,2776514,-3725294,4978142,-6626988
%N A258099 Expansion of ( psi(x^3) * phi(-x^3) / (psi(x) * f(-x^2)) )^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
%C A258099 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258099 G. C. Greubel, <a href="/A258099/b258099.txt">Table of n, a(n) for n = 0..1000</a>
%H A258099 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258099 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258099 Expansion of q^(-1/3) * (eta(q) * eta(q^3) * eta(q^6) / eta(q^2)^3)^2 in powers of q.
%F A258099 Euler transform of period 6 sequence [ -2, 4, -4, 4, -2, 0, ...].
%F A258099 G.f.: Product_{k>0} (1 - x^k + x^(2*k))^2 * (1 + x^k + x^(2*k))^4 / (1 + x^k)^4.
%F A258099 a(n) = A258100(3*n + 1).
%F A258099 a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018
%e A258099 G.f. = 1 - 2*x + 5*x^2 - 12*x^3 + 26*x^4 - 50*x^5 + 92*x^6 - 168*x^7 + ...
%e A258099 G.f. = q - 2*q^4 + 5*q^7 - 12*q^10 + 26*q^13 - 50*q^16 + 92*q^19 + ...
%t A258099 a[ n_] := SeriesCoefficient[ (QPochhammer[ x] * QPochhammer[ x^3] *  QPochhammer[ x^6] / QPochhammer[ x^2]^3)^2, {x, 0, n}];
%o A258099 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x^2 + A)^3)^2, n))};
%Y A258099 Cf. A132977, A258100.
%K A258099 sign
%O A258099 0,2
%A A258099 _Michael Somos_, May 20 2015