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A128143 Expansion of q* (psi(q^9)/phi(q^9))/ (psi(q)/phi(q)) in powers of q where psi(),phi() are Ramanujan theta functions.

%I A128143 #10 Feb 16 2025 08:33:04
%S A128143 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,-7,0,5,0,
%T A128143 -10,-1,12,0,-10,0,14,4,-17,0,21,0,-22,-4,24,0,-34,0,33,-1,-36,0,45,0,
%U A128143 -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,-4
%N A128143 Expansion of q* (psi(q^9)/phi(q^9))/ (psi(q)/phi(q)) in powers of q where psi(),phi() are Ramanujan theta functions.
%C A128143 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 A128143 G. C. Greubel, <a href="/A128143/b128143.txt">Table of n, a(n) for n = 1..1000</a>
%H A128143 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128143 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128143 Euler transform of period 36 sequence [ 1, -2, 1, 0, 1, -2, 1, 0, 0, -2, 1, 0, 1, -2, 1, 0, 1, 0, 1, 0, 1, -2, 1, 0, 1, -2, 0, 0, 1, -2, 1, 0, 1, -2, 1, 0, ...].
%F A128143 Expansion of eta(q^2)^3* eta(q^9)* eta(q^36)^2/ (eta(q)* eta(q^4)^2* eta(q^18)^3) in powers of q.
%F A128143 G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)=v* (1-v+v^2)* (1-u^2)^2 -(1-u*v)^2* (u-v)^2.
%F A128143 a(6*n) = a(6*n+4) = 0.
%F A128143 A092848(n) = a(6*n+2).
%F A128143 A128144(n) = -a(n) if n>0.
%F A128143 A128145(n) = a(n) if n>0.
%t A128143 A128143[n_] := SeriesCoefficient[q*(QPochhammer[q^9]/QPochhammer[q])* (QPochhammer[q^36]/QPochhammer[q^4])^2*(QPochhammer[q^2]/QPochhammer[q^18])^3, {q, 0, n}]; Rest[Table[A128143[n], {n, 0, 1000}]] (* _G. C. Greubel_, Oct 09 2017 *)
%o A128143 (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x^2+A)^3*eta(x^9+A)*eta(x^36+A)^2/ (eta(x+A)*eta(x^4+A)^2*eta(x^18+A)^3), n))}
%K A128143 sign
%O A128143 1,9
%A A128143 _Michael Somos_, Feb 16 2007