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A275372 Expansion of f(-x) * f(-x^2)^4 / phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.

%I A275372 #21 Feb 16 2025 08:33:36
%S A275372 1,-1,-7,6,20,-13,-34,15,53,-25,-91,52,135,-65,-180,82,253,-133,-343,
%T A275372 160,449,-207,-603,306,780,-348,-979,438,1241,-600,-1557,703,1924,
%U A275372 -890,-2375,1115,2910,-1300,-3535,1620,4318,-1993,-5198,2335,6180,-2783,-7420
%N A275372 Expansion of f(-x) * f(-x^2)^4 / phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.
%C A275372 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A275372 G. C. Greubel, <a href="/A275372/b275372.txt">Table of n, a(n) for n = 0..1000</a>
%H A275372 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A275372 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A275372 Expansion of f(-x^2)^7 / (f(x) * f(x^2)^2) in powers of x where f() is a Ramanujan theta function.
%F A275372 Expansion of q^(-3/8) * eta(q) * eta(q^2)^6 * eta(q^8)^2 / eta(q^4)^5 in powers of q.
%F A275372 Euler transform of period 8 sequence [ -1, -7, -1, -2, -1, -7, -1, -4, ...].
%F A275372 G.f.: Product_{k>0} (1 - x^k)^4 * (1 + x^k)^3 * (1 + x^(4*k))^2 / (1 + x^(2*k))^3.
%F A275372 2 * a(n) = - A279955(2*n + 1).
%e A275372 G.f. = 1 - x - 7*x^2 + 6*x^3 + 20*x^4 - 13*x^5 - 34*x^6 + 15*x^7 + 53*x^8 + ...
%e A275372 G.f. = q^3 - q^11 - 7*q^19 + 6*q^27 + 20*q^35 - 13*q^43 - 34*q^51 + 15*q^59 + ...
%t A275372 a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2]^4 / EllipticTheta[ 3, 0, x^2], {x, 0, n}];
%t A275372 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^7 / (QPochhammer[ -x] QPochhammer[ -x^2]^2), {x, 0, n}];
%o A275372 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^6 * eta(x^8 + A)^2 / eta(x^4 + A)^5, n))};
%Y A275372 Cf. A279955.
%K A275372 sign
%O A275372 0,3
%A A275372 _Michael Somos_, Dec 25 2016