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
Keywords
Examples
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 + ...
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- Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17, 4th equation.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.
Programs
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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
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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 *)
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