A053259 Coefficients of the '5th-order' mock theta function phi_1(q).
0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 4, 2, 2, 4, 4, 3, 3, 4, 4, 4, 3, 4, 5, 4, 4, 5, 5, 4, 4, 5, 6, 5, 4, 6, 7, 5, 5, 6, 7, 6, 6, 7, 7, 7, 6, 8, 9, 7, 7, 9
Offset: 0
References
- Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
- Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22, 25.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
- George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
- George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
- George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.
Crossrefs
Programs
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Mathematica
Series[Sum[q^(n+1)^2 Product[1+q^(2k-1), {k, 1, n}], {n, 0, 9}], {q, 0, 100}] nmax = 100; CoefficientList[Series[Sum[x^((k+1)^2) * Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)
Formula
G.f.: phi_1(q) = Sum_{n>=0} q^(n+1)^2 (1+q)(1+q^3)...(1+q^(2n-1)).
a(n) is the number of partitions of n into odd parts such that each occurs at most twice, the largest part is unique and if k occurs as a part then all smaller positive odd numbers occur.
a(n) ~ exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019
Comments