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A053276
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Coefficients of the '7th-order' mock theta function F_1(q).
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6
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0, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 2, 4, 4, 4, 4, 6, 5, 6, 6, 7, 8, 9, 8, 10, 11, 11, 12, 14, 13, 16, 16, 18, 19, 21, 20, 24, 25, 26, 28, 31, 31, 35, 36, 39, 41, 45, 45, 50, 53, 55, 58, 64, 65, 71, 73, 79, 83, 89, 90, 99, 103, 109, 114, 123, 126, 135, 141, 149, 157, 167, 171, 185
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OFFSET
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0,5
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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_1(q) = Sum_{n >= 1} q^n^2/((1-q^n)(1-q^(n+1))...(1-q^(2n-1))).
a(n) = number of partitions of 7n-1 with rank == 2 (mod 7) minus number with rank == 3 (mod 7).
a(n) ~ sin(2*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/Product[1-q^k, {k, n, 2n-1}], {n, 1, 10}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k^2)/Product[1-x^j, {j, k, 2*k-1}], {k, 1, 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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