%I #12 Sep 08 2022 08:45:32

%S 1,9,31,45,6,-45,8,117,121,54,12,-9,14,72,186,261,18,-207,20,270,248,

%T 108,24,63,31,126,391,360,30,-270,32,549,372,162,48,-171,38,180,434,

%U 702,42,-360,44,540,726,216,48,207,57,279,558,630,54,-693,72,936,620

%N Expansion of q * (psi(q^6) / psi(q^3))^3 * phi(q)^5 / psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A133739/b133739.txt">Table of n, a(n) for n = 1..2500</a>

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

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

%F Expansion of eta(q^2)^23 * eta(q^3)^3 * eta(q^12)^6 / (eta(q)^9 * eta(q^4)^10 * eta(q^6)^9) in powers of q.

%F Euler transform of period 12 sequence [ 9, -14, 6, -4, 9, -8, 9, -4, 6, -14, 9, -4, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134078.

%F G.f.: f(x) + 6 * f(x^2) + 27 * f(x^3) + 20 * f(x^4) - 162 * f(x^6) + 108 * f(x^12) where f() is the g.f. of A000203.

%F a(4*n + 2) = 9 * A134077(n). a(6*n + 5) = 6 * A098098(n).

%e G.f. = q + 9*q^2 + 31*q^3 + 45*q^4 + 6*q^5 - 45*q^6 + 8*q^7 + 117*q^8 + ...

%t a[ n_] := SeriesCoefficient[ 2 (EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(3/2)])^3 (EllipticTheta[ 3, 0, x]^5 / EllipticTheta[ 2, 0, x^(1/2)]), {x, 0, n}]; (* _Michael Somos_, Oct 30 2015 *)

%t QP=QPochhammer; CoefficientList[Series[QP[q^2]^23*QP[q^3]^3*QP[q^12]^6/( QP[q]^9*QP[q^4]^10*QP[q^6]^9), {q,0,50}],q] (* _G. C. Greubel_, Nov 16 2018 *)

%o (PARI) {a(n) = my(A); if ( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^23 * eta(x^3 + A)^3 * eta(x^12 + A)^6 / (eta(x + A)^9 * eta(x^4 + A)^10 * eta(x^6 + A)^9), n))};

%o (Magma) A := Basis( ModularForms( Gamma0(12), 2), 58); A[2] + 9*A[3] + 31*A[4] + 45*A[5]; /* _Michael Somos_, Oct 30 2015 */

%Y Cf. A000203, A098098, A134077, A134078.

%K sign

%O 1,2

%A _Michael Somos_, Oct 06 2007