A053277 Coefficients of the '7th-order' mock theta function F_2(q).

1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 4, 5, 4, 6, 5, 7, 7, 8, 8, 10, 9, 11, 11, 13, 13, 16, 15, 17, 18, 21, 20, 23, 23, 27, 27, 31, 31, 35, 35, 39, 41, 45, 45, 51, 51, 57, 59, 64, 66, 73, 74, 81, 83, 91, 93, 102, 104, 113, 117, 126, 130, 141, 144, 156, 162, 174, 178, 192, 198, 212

Offset

0,3

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 Seiichi Manyama)

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_2(q) = Sum_{n >= 0} q^(n(n+1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1)))

a(n) = number of partitions of 7n+2 with rank == 1 (mod 7) minus number with rank == 2 (mod 7).

a(n) ~ sin(3*Pi/7) * exp(Pi*sqrt(2*n/21)) / sqrt(7*n/2). - Vaclav Kotesovec, Jun 15 2019

Mathematica

Series[Sum[q^(n^2+n)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k^2+k)/Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 14 2019 *)

Crossrefs

Other '7th-order' mock theta functions are at A053275, A053276, A053278, A053279, A053280.

Sequence in context: A178697 A255065 A027349 * A078661 A029263 A097575

Adjacent sequences: A053274 A053275 A053276 * A053278 A053279 A053280

Keyword

nonn,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved