A053271 Coefficients of the '6th-order' mock theta function sigma(q).

0, 1, 1, 2, 3, 3, 5, 7, 8, 11, 14, 17, 22, 28, 33, 41, 51, 60, 74, 89, 105, 127, 151, 177, 210, 248, 289, 340, 398, 461, 537, 624, 719, 832, 960, 1101, 1267, 1453, 1660, 1899, 2167, 2465, 2807, 3190, 3614, 4097, 4638, 5237, 5915, 6671, 7507, 8450, 9498

Offset: 0

Author

Dean Hickerson, Dec 19 1999

Keywords

• nonn
• 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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13.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..10000 (terms 0..1000 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, A053272, A053273, A053274.

Programs

Mathematica

Series[Sum[q^((n+1)(n+2)/2) Product[1+q^k, {k, 1, n}]/Product[1-q^k, {k, 1, 2n+1, 2}], {n, 0, 12}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^((k+1)*(k+2)/2) * Product[1+x^j, {j, 1, k}]/Product[1-x^j, {j, 1, 2*k+1, 2}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Formula

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

a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(3*n)). - Vaclav Kotesovec, Jun 12 2019