A053273 Coefficients of the '6th-order' mock theta function 2 mu(q).

1, 2, -3, 4, -4, 6, -11, 14, -15, 22, -31, 34, -41, 56, -69, 82, -98, 120, -152, 178, -204, 254, -308, 354, -415, 496, -587, 680, -785, 922, -1084, 1248, -1427, 1664, -1935, 2202, -2517, 2906, -3336, 3798, -4315, 4930, -5636, 6380, -7202, 8194, -9305, 10474, -11801, 13342, -15050

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13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, A053272, A053274.

Programs

Mathematica

Series[1+Sum[(-1)^n q^(n+1) (1+q^n) Product[1-q^k, {k, 1, 2n-1, 2}]/Product[1+q^k, {k, 1, n+1}], {n, 0, 99}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[1 + Sum[(-1)^k * x^(k+1) * (1+x^k) * Product[1-x^j, {j, 1, 2*k-1, 2}] / Product[1+x^j, {j, 1, k+1}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)

Formula

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

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