A053275 Coefficients of the '7th-order' mock theta function F_0(q).

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

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

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_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

Other '7th-order' mock theta functions are at A053276, A053277, A053278, A053279, A053280.

Sequence in context: A185318 A008622 A029414 * A025816 A025813 A161231

Adjacent sequences: A053272 A053273 A053274 * A053276 A053277 A053278

Keyword

nonn,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved