A053272 Coefficients of the '6th-order' mock theta function lambda(q).

1, -1, 3, -5, 6, -7, 11, -16, 18, -21, 30, -40, 47, -56, 72, -92, 108, -125, 156, -193, 225, -263, 318, -383, 444, -513, 612, -724, 834, -963, 1129, -1320, 1512, -1730, 2010, -2325, 2652, -3022, 3474, -3988, 4524, -5129, 5857, -6673, 7542, -8515, 9660, -10943, 12312, -13842

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (corrected previous b-file from G. C. Greubel)
• George E. Andrews and Dean Hickerson, Ramanujan's "lost" notebook VII: The sixth order mock theta functions, Advances in Mathematics, 89 (1991) 60-105.

Crossrefs

Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053273, A053274.

Programs

Mathematica

Series[Sum[(-q)^n Product[1-q^k, {k, 1, 2n-1, 2}]/Product[1+q^k, {k, 1, n}], {n, 0, 100}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[(-x)^k * Product[1-x^j, {j, 1, 2*k-1, 2}] / Product[1+x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)

Formula

G.f.: lambda(q) = Sum_{n >= 0} (-q)^n (1-q)(1-q^3)...(1-q^(2n-1))/((1+q)(1+q^2)...(1+q^n)).

a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (2*sqrt(3*n)). - Vaclav Kotesovec, Jun 15 2019