A141094 Expansion of b(q) / b(q^2) in powers of q where b() is a cubic AGM theta function.

1, -3, 3, -3, 6, -9, 12, -15, 21, -30, 36, -45, 60, -78, 96, -117, 150, -189, 228, -276, 342, -420, 504, -603, 732, -885, 1050, -1245, 1488, -1773, 2088, -2454, 2901, -3420, 3996, -4662, 5460, -6378, 7404, -8583, 9972, -11565, 13344, -15378, 17748, -20448

Offset: 0

Author

Michael Somos, Jun 04 2008, Aug 12 2009

Keywords

• sign

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).

For n >= 1, a(n)/3 is a weighted count of overpartitions with restricted odd differences. Namely, the number of overpartitions of n counted with weight (-1)^(the largest part) and such that: (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) the smallest part of the overpartition is odd and overlined. - Jeremy Lovejoy, Aug 07 2015

Examples

			G.f. = 1 - 3*q + 3*q^2 - 3*q^3 + 6*q^4 - 9*q^5 + 12*q^6 - 15*q^7 + 21*q^8 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• K. Bringmann, J. Dousse, J. Lovejoy, and K. Mahlburg, Overpartitions with restricted odd differences, Electron. J. Combin. 22 (2015), no.3, paper 3.17.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A092848, A124243, A128128, A132302.

Programs

Maple

with(numtheory):
a:= proc(n) option remember:
      `if`(n=0, 1, add(add(d*[0, -3, 0, -2, 0, -3]
      [irem(d, 6)+1], d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 08 2015

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^3 QPochhammer[ -x^3, x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[n_] := a[n] = If[n==0, 1, Sum[Sum[d{0, -3, 0, -2, 0, -3}[[Mod[d, 6]+1]], {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
a /@ Range[0, 60] (* Jean-François Alcover, Jan 01 2021, after Alois P. Heinz *)

PARI

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

Formula

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

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

Euler transform of period 6 sequence [ -3, 0, -2, 0, -3, 0, ...].

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

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

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 * (u6^2 - u2 * u3) - u6 * (u3^2 - u6*u2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A092848.

a(n) = -3 * A124243(n) unless n=0. a(n) = (-1)^n * A132972(n).

a(2*n) = A128128(n). a(2*n + 1) = - 3* A132302(n).

Convolution inverse of A128128.

Empirical: Sum_{n>=1} exp(-Pi)^(n-1)*(-1)^(n+1)*a(n) = (-2+2*3^(1/2))^(1/3). - Simon Plouffe, Feb 20 2011

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017