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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9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
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved