OFFSET
0,5
COMMENTS
The rank of a partition is its largest part minus the number of parts.
REFERENCES
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.
LINKS
Terry Thibault, Frank Garvan, Table of n, a(n) for n = 0..10000 (terms up to n = 1000 by G. C. Greubel)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
Dean Hickerson, On the seventh order mock theta functions, Inventiones Mathematicae, 94 (1988) 661-677.
FORMULA
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved