A260984 - OEIS

%I #20 Jun 12 2019 17:25:21

%S 1,2,2,2,2,2,4,4,4,4,4,6,6,8,8,8,10,10,12,12,14,16,16,18,20,22,24,26,

%T 28,30,32,36,38,40,44,48,52,56,60,64,68,74,80,84,90,96,104,110,118,

%U 126,134,144,152,162,172,184,196,208,220,234,248

%N Coefficients of the mock theta function chibar(q).

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17, 4th equation.

%H Vaclav Kotesovec, <a href="/A260984/b260984.txt">Table of n, a(n) for n = 0..10000</a>

%H K. Bringmann, J. Dousse, J. Lovejoy, and K. Mahlburg, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p17">Overpartitions with restricted odd differences</a>, Electron. J. Combin. 22 (2015), no.3, paper 3.17.

%H K. Bringmann and J. Lovejoy, <a href="http://arxiv.org/abs/0708.0692">Dyson's rank, overpartitions, and weak Maass forms</a>, arXiv:0708.0692 [math.NT], 2007.

%H K. Bringmann and J. Lovejoy, <a href="https://doi.org/10.1093/imrn/rnm063">Dyson's rank, overpartitions, and weak Maass forms</a>, Int. Math. Res. Not. (2007), rnm063

%F G.f.: 1 + 2*Sum_{n >= 1} q^(n(n+1)/2)*(1+q)^2(1+q^2)^2...(1+q^(n-1))^2*(1+q^n)/((1+q^3)(1+q^6)...(1+q^(3*n)).

%F G.f.: -1 + 2*(1/(1-x) + x^6/((1-x)*(1-x^5)*(1-x^7)) + x^24/((1-x)*(1-x^5)*(1-x^7)*(1-x^11)*(1-x^13) + ...). [Ramanujan] - _Michael Somos_, Sep 13 2016

%F a(n) ~ exp(Pi*sqrt(n)/3) / sqrt(3*n). - _Vaclav Kotesovec_, Jun 12 2019

%e G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 4*x^8 + ...

%p N:= 200: # to get a(0) to a(N)

%p M:= floor((sqrt(1+8*N)-1)/2):

%p G:= 1 + 2*add(q^(n*(n+1)/2)*mul((1+q^i)^2,i=1..n-1)*(1+q^n)/mul(1+q^(3*i),i=1..n), n=1..M):

%p S:= series(G,q,N+1):

%p seq(coeff(S,q,j),j=0..N); # _Robert Israel_, Aug 06 2015

%t a[ n_] := If[ n < 1, Boole[n == 0], 2 SeriesCoefficient[ Sum[ x^(6 k^2) QPochhammer[ x^3, x^6, k] / QPochhammer[ x, x^2, 3 k + 1], {k, 0, Sqrt[n/6]}], {x, 0, n}]]; (* _Michael Somos_, Sep 13 2016 *)

%t nmax = 100; CoefficientList[Series[1 + 2*Sum[x^(k*(k+1)/2) * Product[(1 + x^j), {j, 1, k-1}]^2 * (1 + x^k) / Product[(1 + x^(3*j)), {j, 1, k}], {k, 1, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 12 2019 *)

%K nonn

%O 0,2

%A _Jeremy Lovejoy_, Aug 06 2015