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
Examples
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)
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- Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9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).
Programs
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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 *)
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
Comments