OFFSET
0,2
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17, 4th equation.
LINKS
Vaclav Kotesovec, 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.
K. Bringmann and J. Lovejoy, Dyson's rank, overpartitions, and weak Maass forms, arXiv:0708.0692 [math.NT], 2007.
K. Bringmann and J. Lovejoy, Dyson's rank, overpartitions, and weak Maass forms, Int. Math. Res. Not. (2007), rnm063
FORMULA
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)).
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
a(n) ~ exp(Pi*sqrt(n)/3) / sqrt(3*n). - Vaclav Kotesovec, Jun 12 2019
EXAMPLE
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 + ...
MAPLE
N:= 200: # to get a(0) to a(N)
M:= floor((sqrt(1+8*N)-1)/2):
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):
S:= series(G, q, N+1):
seq(coeff(S, q, j), j=0..N); # Robert Israel, Aug 06 2015
MATHEMATICA
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 *)
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jeremy Lovejoy, Aug 06 2015
STATUS
approved