A053283 Coefficients of the '10th-order' mock theta function X(q).

1, -1, 1, 0, 1, -2, 1, -1, 1, -2, 3, -1, 2, -4, 3, -2, 3, -5, 4, -4, 5, -6, 7, -5, 6, -9, 9, -7, 9, -12, 11, -11, 12, -15, 16, -14, 16, -21, 20, -18, 22, -25, 26, -25, 28, -33, 34, -33, 35, -42, 43, -41, 47, -53, 53, -54, 57, -65, 69, -67, 73, -83, 85, -83, 92, -102, 104, -106, 114, -125, 130, -130, 139, -154

Offset: 0

Author

Dean Hickerson, Dec 19 1999

Keywords

• sign
• easy

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (corrected and extended previous b-file from G. C. Greubel)
• Youn-Seo Choi, Tenth order mock theta functions in Ramanujan's lost notebook, Inventiones Mathematicae, 136 (1999) p. 497-569.

Crossrefs

Other '10th-order' mock theta functions are at A053281, A053282, A053284.

Programs

Mathematica

Series[Sum[(-1)^n q^n^2/Product[1+q^k, {k, 1, 2n}], {n, 0, 10}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[(-1)^k * x^(k^2) / Product[1+x^j, {j, 1, 2*k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)

Formula

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

a(n) ~ (-1)^n * exp(Pi*sqrt(n/10)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019