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Expansion of q^3 * f(q) * f(-q^4) * f(q^15) * f(-q^60) * chi(-q^3) * chi(-q^5) in powers of q where chi(), f() are Ramanujan theta functions.
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%I #12 Feb 16 2025 08:33:24

%S 1,1,-1,-1,-2,-2,0,1,2,0,2,2,0,1,2,0,-4,0,-2,-2,0,1,0,-2,1,-2,-2,0,4,

%T 2,-2,-2,2,1,6,4,0,-1,-8,2,-4,-2,-1,0,-4,0,8,0,4,0,2,-1,2,2,0,2,2,-1,

%U -4,-4,-2,-3,0,2,-8,4,0,-2,4,-2,-4,-6,1,0,0,0,4

%N Expansion of q^3 * f(q) * f(-q^4) * f(q^15) * f(-q^60) * chi(-q^3) * chi(-q^5) in powers of q where chi(), f() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A249371/b249371.txt">Table of n, a(n) for n = 3..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion (1/3) * b(q^2) * c(q^10) * (c(q) * b(q^5) / (b(q) * c(q^5)))^(1/4) in powers of q where b(), c() are cubic AGM theta functions.

%F Expansion of eta(q^2)^3 * eta(q^3) * eta(q^5) * eta(q^30)^3 / (eta(q) * eta(q^6) * eta(q^10) * eta(q^15)) in powers of q.

%F Euler transform of period 30 sequence [ 1, -2, 0, -2, 0, -2, 1, -2, 0, -2, 1, -2, 1, -2, 0, -2, 1, -2, 1, -2, 0, -2, 1, -2, 0, -2, 0, -2, 1, -4, ...].

%e G.f. = q^3 + q^4 - q^5 - q^6 - 2*q^7 - 2*q^8 + q^10 + 2*q^11 + 2*q^13 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[ eta[q^2]^3* eta[q^3]*eta[q^5]*eta[q^30]^3/(eta[q]*eta[q^6]*eta[q^10]*eta[q^15]), {q, 0, 60}], q]] (* _G. C. Greubel_, Aug 13 2018 *)

%o (PARI) {a(n) = my(A); n-=3; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A)^3 / (eta(x + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A)), n))};

%o (Magma) Basis( CuspForms( Gamma0(30), 2), 89) [3];

%K sign,changed

%O 3,5

%A _Michael Somos_, Oct 26 2014