|
|
A053275
|
|
Coefficients of the '7th-order' mock theta function F_0(q).
|
|
6
|
|
|
1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 1, 2, 1, 3, 2, 3, 3, 3, 2, 5, 3, 5, 4, 6, 5, 7, 5, 7, 7, 9, 7, 11, 9, 11, 11, 13, 12, 15, 13, 17, 16, 19, 17, 23, 21, 24, 23, 27, 26, 32, 29, 34, 34, 38, 37, 44, 42, 48, 48, 54, 52, 60, 58, 66, 67, 73, 72, 82, 81, 90, 90, 100, 99, 111, 110, 121, 123
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,8
|
|
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
|
|
|
FORMULA
|
G.f.: F_0(q) = Sum_{n >= 0} q^n^2/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))).
a(n) = number of partitions of 7n with rank == 0 (mod 7) minus number with rank == 2 (mod 7).
a(n) ~ sin(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+1, 2n}], {n, 0, 10}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k^2)/Product[1-x^j, {j, k+1, 2*k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 14 2019 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|