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

0,4

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13.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

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

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 *)

Crossrefs

Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053272, A053273, A053274.

Sequence in context: A081217 A335139 A320036 * A035360 A377435 A027587

Adjacent sequences: A053268 A053269 A053270 * A053272 A053273 A053274

Keyword

nonn,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved