(MAGMAMagma) A := Basis( ModularForms( Gamma0(12), 2), 58); A[2] + 9*A[3] + 31*A[4] + 45*A[5]; /* Michael Somos, Oct 30 2015 */

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

M. Somos, <a href="http://somos.crg4.comA010815/multiqa010815.htmltxt">Introduction to Ramanujan theta functions</a>

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

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 *)

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Expansion of eta(q^2)^23 * eta(q^3)^3 * etapsi(q^12)^6 ) / ( etapsi(q^3))^9 3 * etaphi(q^4)^10 * eta5 / psi(q^6)^9 ) in powers of q where phi(), psi() are Ramanujan theta functions.

1, 9, 31, 45, 6, -45, 8, 117, 121, 54, 12, -9, 14, 72, 186, 261, 18, -207, 20, 270, 248, 108, 24, 63, 31, 126, 391, 360, 30, -270, 32, 549, 372, 162, 48, -171, 38, 180, 434, 702, 42, -360, 44, 540, 726, 216, 48, 207, 57, 279, 558, 630, 54, -693, 72, 936, 620, 270, 60, -54

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).

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

M. Somos, <a href="http://cis.csuohio.edu/~somos.crg4.com/multiq.pdfhtml">Introduction to Ramanujan theta functions</a>

Expansion of q * (psieta(q^62) / psi^23 * eta(q^3))^3 * phieta(q^12)^5 6 / psi(eta(q)^9 * eta(q^4)^10 * eta(q^6)^9) in powers of q where phi(), psi() are Ramanujan theta functions.

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 Pi i t) and g() is the g.f. for A134078.

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.

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

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

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 *)

(PARI) {a(n) = localmy(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))};

(MAGMA) A := Basis( ModularForms( Gamma0(12), 2), 58); A[2] + 9*A[3] + 31*A[4] + 45*A[5]; /* Michael Somos, Oct 30 2015 */

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

Cf. A000203, A098098, A134077, A134078.

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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) (A10054A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

_Michael Somos, _, Oct 06 2007