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

0,2

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9.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

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

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 *)

Crossrefs

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

Sequence in context: A077768 A143038 A029040 * A339395 A228117 A286218

Adjacent sequences: A053278 A053279 A053280 * A053282 A053283 A053284

Keyword

nonn,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved