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A280384 Expansion of f(x)^3 * f(-x^2) * chi(x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.

%I A280384 #8 Feb 16 2025 08:33:39
%S A280384 1,3,-1,-5,8,-1,-28,11,10,-41,41,26,-53,84,21,-101,76,3,-129,99,14,
%T A280384 -190,187,59,-299,263,62,-336,340,27,-459,370,111,-645,518,228,-774,
%U A280384 806,179,-973,882,147,-1233,955,291,-1565,1325,395,-1883,1767,338,-2318,1994
%N A280384 Expansion of f(x)^3 * f(-x^2) * chi(x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.
%C A280384 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A280384 Alois P. Heinz, <a href="/A280384/b280384.txt">Table of n, a(n) for n = 0..10000</a>
%H A280384 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 A280384 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A280384 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A280384 Expansion of q * eta(q^12)^10 * eta(q^36)^6 / (eta(q^6)^3 * eta(q^18)^3 * eta(q^24)^3 * eta(q^72)^3) in powers of q^6.
%F A280384 Euler transform of period 12 sequence [3, -7, 6, -4, 3, -10, 3, -4, 6, -7, 3, -4, ...].
%F A280384 a(n) = (-1)^n * A280328(n).
%F A280384 a(5*n + 1) / a(1) == A187076(n) (mod 5). a(125*n + 21) / a(21) == A187076(n) (mod 25).
%e A280384 G.f. = 1 + 3*x - x^2 - 5*x^3 + 8*x^4 - x^5 - 28*x^6 + 11*x^7 + 10*x^8 + ...
%e A280384 G.f. = q^-1 + 3*q^5 - q^11 - 5*q^17 + 8*q^23 - q^29 - 28*q^35 + 11*q^41 + ...
%t A280384 a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 QPochhammer[ x^2] QPochhammer[ -x^3, x^6]^3, {x, 0, n}];
%o A280384 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^6 + A)^6 / (eta(x + A)^3 * eta(x^3 + A)^3 * eta(x^4 + A)^3 * eta(x^12 + A)^3), n))};
%Y A280384 Cf. A187076, A280328.
%K A280384 sign
%O A280384 0,2
%A A280384 _Michael Somos_, Jan 01 2017