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A128145 Expansion of psi(q^3)* phi(-q^3)* chi^2(-q^3)/( psi(-q)* phi(-q^18)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

%I A128145 #13 Feb 16 2025 08:33:04
%S A128145 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,-7,0,5,
%T A128145 0,-10,-1,12,0,-10,0,14,4,-17,0,21,0,-22,-4,24,0,-34,0,33,-1,-36,0,45,
%U A128145 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 A128145 Expansion of psi(q^3)* phi(-q^3)* chi^2(-q^3)/( psi(-q)* phi(-q^18)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
%C A128145 Ramanujan theta functions: f(q) := Product_{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) := Product_{k>=0} (1 + q^(2k+1)) (A000700).
%H A128145 G. C. Greubel, <a href="/A128145/b128145.txt">Table of n, a(n) for n = 0..1000</a>
%H A128145 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128145 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128145 Expansion of (eta(q^2)* eta(q^3)^3* eta(q^36))/(eta(q)* eta(q^4)* eta(q^6)* eta(q^18)^2) in powers of q.
%F A128145 Euler transform of period 36 sequence [ 1, 0, -2, 1, 1, -2, 1, 1, -2, 0, 1, -1, 1, 0, -2, 1, 1, 0, 1, 1, -2, 0, 1, -1, 1, 0, -2, 1, 1, -2, 1, 1, -2, 0, 1, 0, ...].
%F A128145 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - 1)*(3 - 3*v + v^2)*(2*u - u^2)^2 - (u + v - u*v)^2*(u - v)^2.
%F A128145 a(6n+4)=0. a(6n)=0 if n > 0.
%t A128145 eta[x_] := x^(1/24)*QPochhammer[x]; A128145[n_] := SeriesCoefficient[ eta[q^2]*eta[q^3]^3*eta[q^36]/(eta[q]*eta[q^4]*eta[q^6]*eta[q^18]^2 ), {q, 0, n}]; Table[A128145[n], {n,0,50}] (* _G. C. Greubel_, Aug 16 2017 *)
%o A128145 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^36+A)*eta(x^3+A)^3/ (eta(x+A)*eta(x^4+A)*eta(x^6+A)*eta(x^18+A)^2), n))}
%Y A128145 A092848(n) = a(6n+2). A128143(n) = a(n) if n > 0. A128144(n) = -a(n) if n > 0.
%K A128145 sign
%O A128145 0,10
%A A128145 _Michael Somos_, Feb 16 2007