A053270 Coefficients of the '6th-order' mock theta function rho(q).

1, 2, 3, 4, 6, 8, 11, 14, 18, 24, 30, 38, 47, 58, 72, 88, 108, 130, 156, 188, 225, 268, 318, 376, 444, 522, 612, 716, 834, 972, 1129, 1308, 1512, 1744, 2010, 2310, 2652, 3038, 3474, 3968, 4524, 5152, 5857, 6650, 7542, 8540, 9660, 10912, 12312, 13878

Offset

0,2

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 3, 13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.: rho(q) = Sum_{n >= 0} ( q^(n(n+1)/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)) / (2*sqrt(3*n)). - Vaclav Kotesovec, Jun 12 2019

Mathematica

Series[Sum[q^(n(n+1)/2) Product[1+q^k, {k, 1, n}]/Product[1-q^k, {k, 1, 2n+1, 2}], {n, 0, 13}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/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, A053271, A053272, A053273, A053274.

Sequence in context: A143611 A279075 A062464 * A261154 A233693 A003412

Adjacent sequences: A053267 A053268 A053269 * A053271 A053272 A053273

Keyword

nonn,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved