A128642 -id:A128642 - cp's OEIS frontend

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

Michael Somos, Oct 03 2006, Jan 21 2009

sign

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

			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 + ...

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

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

M := Basis(ModularForms(Gamma1(6), 1), 43); M1 := M[1]; M2 := M[2]; A := M2/(M1 + 2*M2); A; /* Michael Somos, May 26 2023 */

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 *)

{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))};

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

A128643 Expansion of (b(q^2) / b(q))^3 in powers of q where b() is a cubic AGM function.

Original entry on oeis.org

1, 9, 45, 171, 549, 1566, 4095, 10008, 23157, 51201, 108918, 224100, 447831, 872118, 1659672, 3093498, 5658453, 10173762, 18006021, 31408092, 54053190, 91869192, 154331028, 256447080, 421789671, 687086127, 1109128014, 1775103507

Offset: 0

Michael Somos, Mar 16 2007

nonn

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

			1 + 9*q + 45*q^2 + 171*q^3 + 549*q^4 + 1566*q^5 + 4095*q^6 + 10008*q^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A105559, A128638, A128642, A193522.

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3, q^6] / QPochhammer[ q, q^2]^3)^3, {q, 0, n}]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^2 + A) / eta(x + A))^3 * eta(x^3 + A) / eta(x^6 + A))^3, n))}

Expansion of (chi(-q^3) / chi(-q)^3)^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of ((eta(q^2) / eta(q))^3 * (eta(q^3) / eta(q^6)))^3 in powers of q.
Euler transform of period 6 sequence [ 9, 0, 6, 0, 9, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 - v) * (1 + 8*u) + (u - v)^2.
G.f.: (Product_{k>0} (1 + x^k) / (1 + x^(3*k))^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 g.f. for A105559.
a(n) = 9 * A128638(n) unless n = 0. -4*a(n) = A193522(2*n). Convolution inverse of A128642.
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (8 * 2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 27 2019
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 1/4 + (1/8)*sqrt(3) + (1/8)*sqrt(9+6*sqrt(3)). - Simon Plouffe, Mar 04 2021

cp's OEIS Frontend

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

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