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Expansion of chi(-q)* chi(-q^2)* chi(-q^9)/( chi(-q^3)* chi(q^9)) in powers of q where chi() is a Ramanujan theta function.
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%I #10 Feb 16 2025 08:33:04

%S 1,-1,-1,1,0,-1,0,1,1,-2,0,3,0,-2,0,3,0,-5,0,4,-2,-4,0,5,0,-7,2,7,0,

%T -5,0,10,1,-12,0,10,0,-14,-4,17,0,-21,0,22,4,-24,0,34,0,-33,1,36,0,

%U -45,0,45,-8,-52,0,55,0,-62,8,71,0,-70,0,88,2,-96,0,98,0,-122,-14,133,0,-148,0,163,14,-182,0,217,0,-216

%N Expansion of chi(-q)* chi(-q^2)* chi(-q^9)/( chi(-q^3)* chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

%H G. C. Greubel, <a href="/A128144/b128144.txt">Table of n, a(n) for n = 0..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 of (eta(q)* eta(q^6)* eta(q^36)* eta(q^9)^2)/(eta(q^3)* eta(q^4)* eta(q^18)^3) in powers of q.

%F Euler transform of period 36 sequence [ -1, -1, 0, 0, -1, -1, -1, 0, -2, -1, -1, 0, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, 0, -1, -1, -2, 0, -1, -1, -1, 0, 0, -1, -1, 0, ...].

%F G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)= (1-v)*(1-v+v^2)*(2*u-u^2)^2 -(u+v-u*v)^2*(u-v)^2.

%F a(6*n+4)=0. a(6*n)=0 if n>0.

%F A092848(n) = -a(6*n+2).

%F A128143(n) = -a(n) if n>0.

%F A128145(n) = -a(n) if n>0.

%t A128144[n_] := SeriesCoefficient[((QPochhammer[q]*QPochhammer[q^6] *QPochhammer[q^36]*QPochhammer[q^9]^2)/(QPochhammer[q^3]*QPochhammer[q^4] *QPochhammer[q^18]^3)), {q, 0, n}]; Table[A128144[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 09 2017 *)

%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^6+A)*eta(x^36+A)*eta(x^9+A)^2/ (eta(x^3+A)*eta(x^4+A)*eta(x^18+A)^3), n))}

%K sign,changed

%O 0,10

%A _Michael Somos_, Feb 16 2007