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A164271 Expansion of ( f(-q^2) * f(q^3) * f(-q^6) / f(q)^3 )^2 in powers of q where f() is a Ramanujan theta function.

%I A164271 #12 Feb 16 2025 08:33:11
%S A164271 1,-6,25,-84,248,-666,1662,-3912,8774,-18894,39289,-79248,155612,
%T A164271 -298338,559812,-1030224,1862647,-3313494,5807096,-10037796,17129888,
%U A164271 -28886052,48170178,-79492824,129900206,-210314976,337545438,-537278124,848509124
%N A164271 Expansion of ( f(-q^2) * f(q^3) * f(-q^6) / f(q)^3 )^2 in powers of q where f() is a Ramanujan theta function.
%C A164271 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A164271 G. C. Greubel, <a href="/A164271/b164271.txt">Table of n, a(n) for n = 0..1000</a>
%H A164271 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A164271 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A164271 Expansion of ( chi(q) * phi(q^3) * psi(-q^3) / phi(q^2)^2 )^2 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
%F A164271 Euler transform of period 12 sequence [ -6, 10, -4, 4, -6, 4, -6, 4, -4, 10, -6, 0, ...].
%F A164271 G.f.: (Product_{k>0} (1 - x^k + x^(2*k))^3 * (1 + x^(2*k))^2 * (1 + x^k + x^(2*k))^2 / ((1 + x^k)^2 * (1 - x^(2*k) + x^(4*k))))^2.
%F A164271 2 * a(n) = A164269(3*n + 2).
%F A164271 Expansion of ( f(-x^6)^6 / (f(x, x^5) * f(x)^2) )^2 in powers of x where f(,) is Ramanujan's general theta function. - _Michael Somos_, Sep 02 2015
%e A164271 G.f. = 1 - 6*x + 25*x^2 - 84*x^3 + 248*x^4 - 666*x^5 + 1662*x^6 - 3912*x^8 + ...
%e A164271 G.f. = q^2 - 6*q^5 + 25*q^8 - 84*q^11 + 248*q^14 - 666*q^17 + 1662*q^20 + ...
%t A164271 f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164271[n_] := SeriesCoefficient[((f[-q^2, -q^4]*f[q^3, -q^6]*f[-q^6, -q^12])/f[q, -q^2]^3)^2, {q, 0, n}]; Table[A164271[n], {n,0,50}] (* _G. C. Greubel_, Sep 16 2017 *)
%o A164271 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^4 / (eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A)))^2, n))};
%Y A164271 Cf. A164269.
%K A164271 sign
%O A164271 0,2
%A A164271 _Michael Somos_, Aug 11 2009