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A141094 Expansion of b(q) / b(q^2) in powers of q where b() is a cubic AGM function. 6
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, 23472, -26910 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

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

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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 g.f. for A092848.

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

EXAMPLE

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

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

PROG

(PARI) {a(n) = local(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))}

CROSSREFS

-3 * A124243(n) = a(n) unless n=0. -(-1)^n * A132972(n) = a(n). A128128(n) = a(2*n). -3 * A132302(n) = a(2*n + 1).

Convolution inverse of A128128.

Sequence in context: A153004 A214361 A124449 * A132972 A113920 A081848

Adjacent sequences:  A141091 A141092 A141093 * A141095 A141096 A141097

KEYWORD

sign

AUTHOR

Michael Somos, Jun 04 2008, Aug 12 2009

STATUS

approved

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Last modified September 1 18:17 EDT 2015. Contains 261279 sequences.