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A280328 Expansion of f(-x)^3 * f(-x^2) * chi(-x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.

%I A280328 #11 Feb 16 2025 08:33:38
%S A280328 1,-3,-1,5,8,1,-28,-11,10,41,41,-26,-53,-84,21,101,76,-3,-129,-99,14,
%T A280328 190,187,-59,-299,-263,62,336,340,-27,-459,-370,111,645,518,-228,-774,
%U A280328 -806,179,973,882,-147,-1233,-955,291,1565,1325,-395,-1883,-1767,338,2318
%N A280328 Expansion of f(-x)^3 * f(-x^2) * chi(-x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.
%C A280328 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A280328 G. C. Greubel, <a href="/A280328/b280328.txt">Table of n, a(n) for n = 0..2500</a>
%H A280328 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 A280328 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A280328 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A280328 Expansion of q * eta(q^6)^3 * eta(q^12) * eta(q^18)^3 / eta(q^36)^3 in powers of q^6.
%F A280328 Euler transform of period 6 sequence [-3, -4, -6, -4, -3, -4, ...].
%F A280328 a(n) = (-1)^n * A280384(n).
%F A280328 a(5*n + 1) / a(1) == A000727(n) (mod 5). a(125*n + 21) / a(21) == A000727(n) (mod 25).
%e A280328 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 A280328 G.f. = q^-1 - 3*q^5 - q^11 + 5*q^17 + 8*q^23 + q^29 - 28*q^35 - 11*q^41 + ...
%t A280328 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 QPochhammer[ x^2] QPochhammer[ x^3, x^6]^3, {x, 0, n}];
%o A280328 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^2 + A) * eta(x^3 + A)^3 / eta(x^6 + A)^3, n))};
%Y A280328 Cf.A000727, A280384.
%K A280328 sign
%O A280328 0,2
%A A280328 _Michael Somos_, Dec 31 2016