A053282 Coefficients of the '10th-order' mock theta function psi(q).

0, 1, 1, 2, 2, 2, 4, 4, 4, 6, 7, 8, 10, 11, 12, 16, 18, 20, 24, 26, 30, 36, 40, 44, 52, 58, 64, 74, 82, 91, 104, 116, 128, 144, 159, 176, 198, 218, 240, 268, 294, 324, 360, 394, 432, 478, 524, 572, 630, 688, 752, 826, 900, 980, 1072, 1168, 1270, 1386, 1505, 1634

Offset

0,4

Comments

Number of partitions (d1,d2,...,dm) of n such that 0 < d1/1 <= d2/2 <= ... <= dm/m. - Seiichi Manyama, Mar 17 2018

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9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

Michele Nardelli, Antonio Nardelli, On the Ramanujan's Mock theta functions of tenth order: new possible mathematical developments and mathematical connections with some sectors of Particle Physics and Black Hole physics II, Università degli Studi di Napoli (Italy, 2019).

Formula

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

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

Example

From Seiichi Manyama, Mar 17 2018: (Start)
n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m)
--+--------------------------+-------------------------
1 | (1)                      | (1)
2 | (2)                      | (2)
3 | (3)                      | (3)
  | (1, 2)                   | (1, 1)
4 | (4)                      | (4)
  | (1, 3)                   | (1, 3/2)
5 | (5)                      | (5)
  | (1, 4)                   | (1, 2)
6 | (6)                      | (6)
  | (1, 5)                   | (1, 5/2)
  | (2, 4)                   | (2, 2)
  | (1, 2, 3)                | (1, 1, 1)
7 | (7)                      | (7)
  | (1, 6)                   | (1, 3)
  | (2, 5)                   | (2, 5/2)
  | (1, 2, 4)                | (1, 1, 4/3)
8 | (8)                      | (8)
  | (1, 7)                   | (1, 7/2)
  | (2, 6)                   | (2, 3)
  | (1, 2, 5)                | (1, 1, 5/3)
9 | (9)                      | (9)
  | (1, 8)                   | (1, 4)
  | (2, 7)                   | (2, 7/2)
  | (3, 6)                   | (3, 3)
  | (1, 2, 6)                | (1, 1, 2)
  | (1, 3, 5)                | (1, 3/2, 5/3) (End)

Mathematica

Series[Sum[q^((n+1)(n+2)/2)/Product[1-q^(2k+1), {k, 0, n}], {n, 0, 12}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^((k+1)*(k+2)/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 A053281, A053283, A053284.

Sequence in context: A086160 A029047 A007294 * A218084 A240046 A363947

Adjacent sequences: A053279 A053280 A053281 * A053283 A053284 A053285

Keyword

nonn,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved