A053265 Coefficients of the '5th-order' mock theta function F_1(q).

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 20, 21, 24, 26, 28, 31, 34, 37, 40, 44, 47, 51, 56, 60, 65, 71, 76, 82, 89, 95, 103, 111, 119, 128, 138, 148, 158, 171, 182, 195, 210, 223, 239, 256, 273, 292, 312, 332, 354, 378, 402, 428

Offset

0,5

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 22, 25.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

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.

Formula

G.f.: F_1(q) = Sum_{n>=0} q^(2n(n+1))/((1-q)(1-q^3)...(1-q^(2n+1))).

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

Mathematica

Series[Sum[q^(2n(n+1))/Product[1-q^(2k+1), {k, 0, n}], {n, 0, 6}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(2*k*(k+1)) / Product[1-x^(2*j+1), {j, 0, k}], {k, 0, Floor[Sqrt[nmax/2]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053266, A053267.

Sequence in context: A189709 A025771 A154403 * A035452 A199575 A120187

Adjacent sequences: A053262 A053263 A053264 * A053266 A053267 A053268

Keyword

nonn,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved