A123633 Expansion of (c(q^2)/c(q))^3 in powers of q where c() is a cubic AGM theta function.

1, -3, 3, 5, -18, 15, 24, -75, 57, 86, -252, 183, 262, -744, 522, 725, -1998, 1365, 1852, -4986, 3336, 4436, -11736, 7719, 10103, -26322, 17067, 22040, -56682, 36306, 46336, -117867, 74700, 94378, -237744, 149277, 186926, -466836, 290706, 361126, -895014, 553224

Offset

1,2

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).

In the arXiv:2305.13951 paper on page 21 is this: "The q-expansion of y coincides with the sequence A123633 in the OEIS". - Michael Somos, May 26 2023

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Xuhang Jiang, Xing Wang, Li Lin Yang and Jing-Bang Zhao, Epsilon-factorized differential equations for two-loop non-planar triangle Feynman integrals with elliptic curves, arXiv:2305.13951 [hep-th], 2023. See page 21.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q / (chi(-q^3)^3 / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.

Euler transform of period 6 sequence [ -3, 0, 6, 0, -3, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v)= u^2 - v - u*v * (6 + 8*v).

G.f.: x * (Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3 )^3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1 / 8) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128642.

A128636(n) = a(n) unless n = 0. Convolution inverse of A105559.

Convolution cube of A092848.

Convolution with A123330 is A093829. - Michael Somos, May 26 2023

Example

G.f. = q - 3*q^2 + 3*q^3 + 5*q^4 - 18*q^5 + 15*q^6 + 24*q^7 - 75*q^8 + 57*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q / (QPochhammer[ q^3, q^6]^3 / QPochhammer[ q, q^2])^3, {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 1, n, 2}] / Product[ 1 - q^k, {k, 3, n, 6}]^3)^3, {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^3 * (eta(x^6 + A) / eta(x^3 + A))^9, n))};
(Magma) M := Basis(ModularForms(Gamma1(6), 1), 43); M1 := M[1]; M2 := M[2]; A<q> := M2/(M1 + 2*M2); A; /* Michael Somos, May 26 2023 */

Crossrefs

Cf. A092848, A093829, A105559, A123330, A128636, A128642.

Sequence in context: A369345 A144423 A128636 * A215912 A157241 A365083

Adjacent sequences: A123630 A123631 A123632 * A123634 A123635 A123636

Keyword

sign

Author

Michael Somos, Oct 03 2006, Jan 21 2009

Status

approved