A053252 Coefficients of the '3rd-order' mock theta function chi(q).

1, 1, 1, 0, 0, 0, 1, 1, 0, 0, -1, 0, 1, 1, 1, -1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 1, 1, 0, -1, -1, 1, 1, 0, -1, -1, 0, 1, 2, 1, -1, -1, 0, 1, 1, 0, -1, -2, 0, 1, 2, 1, -1, -1, -1, 1, 2, 1, -1, -2, -1, 2, 2, 1, -1, -2, -1, 1, 2, 0, -1, -3, 0, 2, 3, 2, -2, -2, -1, 2, 3, 0, -2, -3, -1, 2, 3, 2, -3, -3, -1, 2, 4, 1, -2, -4, -1, 3, 4, 2, -2, -4

Offset: 0

Author

Dean Hickerson, Dec 19 1999

Keywords

• sign
• easy

References

• N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.14).
• Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
• John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015.
• George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Crossrefs

Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053253, A053254, A053255, A261401.

Programs

Mathematica

Series[Sum[q^n^2/Product[1-q^k+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]

Formula

G.f.: chi(q) = Sum_{n >= 0} q^n^2/((1-q+q^2)*(1-q^2+q^4)*...*(1-q^n+q^(2n))).

G.f.: G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)) / G(k+1). - Joerg Arndt, Jun 29 2013