A053268 Coefficients of the '6th-order' mock theta function phi(q).

1, -1, 2, -1, 1, -3, 3, -3, 4, -4, 6, -6, 5, -9, 11, -10, 11, -15, 17, -16, 19, -22, 26, -29, 29, -36, 42, -42, 46, -55, 60, -64, 71, -79, 90, -95, 101, -117, 131, -137, 148, -169, 184, -195, 211, -234, 258, -276, 295, -327, 360, -379, 409, -453, 489, -522, 563, -612, 666, -710, 757, -829, 898

Offset

0,3

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 2, 4, 6, 13, 16, 17

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (corrected and extended 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.

Formula

G.f.: phi(q) = sum for n >= 0 of (-1)^n q^n^2 (1-q)(1-q^3)...(1-q^(2n-1))/((1+q)(1+q^2)...(1+q^(2n))).

a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(3*n)). - Vaclav Kotesovec, Jun 15 2019

Mathematica

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

Crossrefs

Other '6th-order' mock theta functions are at A053269, A053270, A053271, A053272, A053273, A053274.

Sequence in context: A355201 A334347 A283672 * A284828 A101417 A318660

Adjacent sequences: A053265 A053266 A053267 * A053269 A053270 A053271

Keyword

sign,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved