A053281 Coefficients of the '10th-order' mock theta function phi(q).

1, 2, 2, 3, 4, 4, 6, 7, 8, 10, 12, 14, 16, 20, 22, 26, 31, 34, 40, 46, 52, 60, 68, 76, 87, 98, 110, 124, 140, 156, 174, 196, 216, 242, 270, 298, 332, 368, 406, 449, 496, 546, 602, 664, 728, 800, 880, 962, 1056, 1156, 1262, 1381, 1508, 1644, 1794, 1956, 2128

Offset: 0

Author

Dean Hickerson, Dec 19 1999

Keywords

• nonn
• easy

Comments

The alternating sum of the same series, namely phi(q) = Sum_{n>=0} (-1)^n q^(n(n+1)/2)/((1-q)(1-q^3)...(1-q^(2n+1))) = 1 + x^3 - x^7 - x^16 + x^24 + x^39 - x^51 - ..., where the exponents are given by 5n^2 +- 2n. See the Amer. Math. Monthly reference.

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
• Youn-Seo Choi, Tenth order mock theta functions in Ramanujan's lost notebook, Inventiones Mathematicae, 136 (1999) p. 497-569.
• David Newman, A Recurrence inside a Generating Function: Solution to problem 10681, American Mathematical Monthly, vol. 107 (2000), p. 569.

Crossrefs

Other '10th-order' mock theta functions are at A053282, A053283, A053284.

Programs

Mathematica

Series[Sum[q^(n(n+1)/2)/Product[1-q^(2k+1), {k, 0, n}], {n, 0, 13}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2) / Product[1-x^(2*j+1), {j, 0, k}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)

Formula

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

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