A187153 Expansion of q * (psi(q) / psi(q^2)) / (psi(q^3) / psi(q^6))^3 in powers of q where psi() is a Ramanujan theta function.

1, 1, -1, -3, -2, 3, 8, 5, -7, -18, -12, 15, 38, 24, -30, -75, -46, 57, 140, 86, -104, -252, -152, 183, 439, 262, -313, -744, -442, 522, 1232, 725, -852, -1998, -1168, 1365, 3182, 1852, -2150, -4986, -2886, 3336, 7700, 4436, -5106, -11736, -6736, 7719

Offset

1,4

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

The Alaca, et. al. paper in equation (1.8) on page 178 uses the notation p(q) := (phi^2(q) - phi^2(q^3)/(2 phi^2(q^3)), where phi(q) is a Ramanujan theta function. This p(q) is twice the g.f. of this sequence. - Michael Somos, Mar 30 2021

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

A. Alaca, S. Alaca, K. S. Williams, On the two-dimensional theta functions of the Borweins

Johannes Blümlein, Iterative Non-iterative Integrals in Quantum Field Theory, arXiv:1808.08128 [hep-th], 2018.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

W. Zudilin, Deciding whether a given power series is modular or not

Formula

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

Euler transform of period 12 sequence [1, -2, -2, 0, 1, 4, 1, 0, -2, -2, 1, 0, ...].

Expansion of c(q) * c(q^4)^2 / c(q^2)^3 in powers of q where c() is a cubic AGM theta function.

If p = 2 * A(q), then B(q) = p * ((2 + p) / (1 + 2*p))^3 and B(q^3) = p^3 * ((2 + p) / (1 + 2*p)) where B() is the g.f. for A115977. - Michael Somos, Feb 27 2012

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

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^2 * (1 + 4*v) * (3*v + u*(1 + 4*v)) - v * (1 + v) * (3*u + 1 + v).

Convolution inverse of A187143.

Expansion of (phi^2(q) - phi^2(q^3)/(4*phi^2(q^3)), where phi(q) is a Ramanujan theta function. - Michael Somos, Mar 30 2021

Example

G.f. = q + q^2 - q^3 - 3*q^4 - 2*q^5 + 3*q^6 + 8*q^7 + 5*q^8 - 7*q^9 - 18*q^10 + ...

Mathematica

QP = QPochhammer; s = (QP[q^2]*QP[q^3]*QP[q^12]^2)^3/(QP[q]*QP[q^4]^2* QP[q^6]^9) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
QP = QPochhammer; Rest[Table[SeriesCoefficient[q*(QP[-q, q^2]*QP[-q^6, q^6]^3)/(QP[-q^2, q^2]*QP[-q^3, q^6]^3), {q, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Dec 04 2017 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ x^2, x^4]^2 QPochhammer[x^3, x^6]^3 QPochhammer[ -x^6, x^6]^6, {x, 0, n}]; (* Michael Somos, Oct 15 2018 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, x]^2 / EllipticTheta[3, 0, x^3]^2 - 1)/4, {x, 0, n}]; (* Michael Somos, Mar 30 2021 *)

Prog

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

Crossrefs

Cf. A115977, A123649, A187143, A213265.

Sequence in context: A331926 A070982 A275520 * A213265 A123649 A080848

Adjacent sequences: A187150 A187151 A187152 * A187154 A187155 A187156

Keyword

sign

Author

Michael Somos, Mar 06 2011

Status

approved