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 A053282 Coefficients of the '10th-order' mock theta function psi(q). 10
 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 (list; graph; refs; listen; history; text; internal format)
 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= dm/m. - Seiichi Manyama, Mar 17 2018 REFERENCES Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama) Youn-Seo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 A001584 Adjacent sequences:  A053279 A053280 A053281 * A053283 A053284 A053285 KEYWORD nonn,easy AUTHOR Dean Hickerson, Dec 19 1999 STATUS approved

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Last modified June 17 00:17 EDT 2021. Contains 345080 sequences. (Running on oeis4.)