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A279955 Expansion of chi(-x^4)^4 * f(-x^4)^2 * f(-x)^2 in powers of x where chi(), f() are Ramanujan theta functions.

%I A279955 #11 Feb 16 2025 08:33:38
%S A279955 1,-2,-1,2,-5,14,4,-12,5,-40,0,26,11,68,-15,-30,-18,-106,3,50,-10,182,
%T A279955 29,-104,10,-270,11,130,37,360,-51,-164,-16,-506,-30,266,-65,686,62,
%U A279955 -320,53,-898,22,414,50,1206,-61,-612,-52,-1560,-4,696,-81,1958,120
%N A279955 Expansion of chi(-x^4)^4 * f(-x^4)^2 * f(-x)^2 in powers of x where chi(), f() are Ramanujan theta functions.
%C A279955 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A279955 G. C. Greubel, <a href="/A279955/b279955.txt">Table of n, a(n) for n = 0..1000</a>
%H A279955 Amanda Clemm, <a href="http://www.mdpi.com/2227-7390/4/1/5">Modular Forms and Weierstrass Mock Modular Forms</a>, Mathematics, volume 4, issue 1, (2016)
%H A279955 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A279955 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A279955 Expansion of q * eta(q^4)^2 * eta(q^16)^6 / eta(q^32)^4 in powers of q^4.
%F A279955 Euler transform of period 8 sequence [ -2, -2, -2, -8, -2, -2, -2, -4, ...].
%F A279955 a(n) = (-1)^n * A280339(n).
%F A279955 a(3*n + 1) / a(1) == A002171(n) (mod 3). a(3^3*n + 7) / a(7) == A002171(n) (mod 3^2).
%e A279955 G.f. = 1 - 2*x - x^2 + 2*x^3 - 5*x^4 + 14*x^5 + 4*x^6 - 12*x^7 + 5*x^8 + ...
%e A279955 G.f. = q^-1 - 2*q^3 - q^7 + 2*q^11 - 5*q^15 + 14*q^19 + 4*q^23 - 12*q^27 + ...
%t A279955 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^4]^2 QPochhammer[ x^4, x^8]^4, {x, 0, n}];
%o A279955 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^6 / eta(x^8 + A)^4, n))};
%Y A279955 Cf. A002171, A280339.
%K A279955 sign
%O A279955 0,2
%A A279955 _Michael Somos_, Dec 23 2016