A141094 - OEIS

%I #50 Feb 16 2025 08:33:08

%S 1,-3,3,-3,6,-9,12,-15,21,-30,36,-45,60,-78,96,-117,150,-189,228,-276,

%T 342,-420,504,-603,732,-885,1050,-1245,1488,-1773,2088,-2454,2901,

%U -3420,3996,-4662,5460,-6378,7404,-8583,9972,-11565,13344,-15378,17748,-20448

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

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

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

%C 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

%H Alois P. Heinz, <a href="/A141094/b141094.txt">Table of n, a(n) for n = 0..10000</a>

%H K. Bringmann, J. Dousse, J. Lovejoy, and K. Mahlburg, <a href="https://doi.org/10.37236/5248">Overpartitions with restricted odd differences</a>, Electron. J. Combin. 22 (2015), no.3, paper 3.17.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

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

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

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

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

%F 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).

%F 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).

%F 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.

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

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

%F Convolution inverse of A128128.

%F 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

%F a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017

%e 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 + ...

%p with(numtheory):

%p a:= proc(n) option remember:

%p       `if`(n=0, 1, add(add(d*[0, -3, 0, -2, 0, -3]

%p       [irem(d, 6)+1], d=divisors(j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Aug 08 2015

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^3 QPochhammer[ -x^3, x^3], {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)

%t 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];

%t a /@ Range[0, 60] (* _Jean-François Alcover_, Jan 01 2021, after _Alois P. Heinz_ *)

%o (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))};

%Y Cf. A092848, A124243, A128128, A132302.

%K sign

%O 0,2

%A _Michael Somos_, Jun 04 2008, Aug 12 2009