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A053277
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Coefficients of the '7th-order' mock theta function F_2(q).
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6
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1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 4, 5, 4, 6, 5, 7, 7, 8, 8, 10, 9, 11, 11, 13, 13, 16, 15, 17, 18, 21, 20, 23, 23, 27, 27, 31, 31, 35, 35, 39, 41, 45, 45, 51, 51, 57, 59, 64, 66, 73, 74, 81, 83, 91, 93, 102, 104, 113, 117, 126, 130, 141, 144, 156, 162, 174, 178, 192, 198, 212
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OFFSET
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0,3
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COMMENTS
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The rank of a partition is its largest part minus the number of parts.
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REFERENCES
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Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Atle Selberg, Uber die Mock-Thetafunktionen siebenter Ordnung, Arch. Math. Naturvidenskab, 41 (1938) 3-15.
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LINKS
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FORMULA
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G.f.: F_2(q) = Sum_{n >= 0} q^(n(n+1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1)))
a(n) = number of partitions of 7n+2 with rank == 1 (mod 7) minus number with rank == 2 (mod 7).
a(n) ~ sin(3*Pi/7) * exp(Pi*sqrt(2*n/21)) / sqrt(7*n/2). - Vaclav Kotesovec, Jun 15 2019
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MATHEMATICA
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Series[Sum[q^(n^2+n)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k^2+k)/Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 14 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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