A053264 Coefficients of the '5th-order' mock theta function F_0(q).

1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6, 6, 7, 8, 8, 10, 11, 11, 13, 14, 15, 17, 18, 19, 22, 24, 25, 28, 30, 32, 36, 39, 41, 45, 49, 52, 57, 61, 65, 71, 76, 81, 88, 94, 100, 109, 116, 123, 133, 142, 151, 163, 174, 184, 198, 211, 224, 240, 255, 271, 290

Offset: 0

Author

Dean Hickerson, Dec 19 1999

Keywords

• nonn
• easy

References

• Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 22, 23, 25.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
• George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
• George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
• George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053265, A053266, A053267.

Programs

Mathematica

Series[Sum[q^(2n^2)/Product[1-q^(2k+1), {k, 0, n-1}], {n, 0, 7}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(2*k^2) / Product[1-x^(2*j+1), {j, 0, k-1}], {k, 0, Floor[Sqrt[nmax/2]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Formula

G.f.: F_0(q) = Sum_{n>=0} q^(2n^2)/((1-q)(1-q^3)...(1-q^(2n-1))).

a(n) is the number of partitions of n into odd parts, each of which occurs at least twice, such that if k occurs then all smaller positive odd numbers occur.

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(3/2)*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019