A260984 Coefficients of the mock theta function chibar(q).

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, 28, 30, 32, 36, 38, 40, 44, 48, 52, 56, 60, 64, 68, 74, 80, 84, 90, 96, 104, 110, 118, 126, 134, 144, 152, 162, 172, 184, 196, 208, 220, 234, 248

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17, 4th equation.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

Sequence in context: A001300 A169718 A001306 * A108105 A321213 A063468

Adjacent sequences: A260981 A260982 A260983 * A260985 A260986 A260987

Keyword

nonn

Author

Jeremy Lovejoy, Aug 06 2015

Status

approved