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A053282
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Coefficients of the '10th-order' mock theta function psi(q).
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10
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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
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OFFSET
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0,4
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COMMENTS
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Number of partitions (d1,d2,...,dm) of n such that 0 < d1/1 <= d2/2 <= ... <= dm/m. - Seiichi Manyama, Mar 17 2018
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REFERENCES
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9
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LINKS
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FORMULA
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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
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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