A053262 Coefficients of the 5th-order mock theta function chi_0(q).

1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 7, 9, 7, 12, 11, 13, 13, 17, 15, 21, 20, 24, 24, 29, 28, 36, 35, 40, 42, 50, 48, 58, 58, 67, 70, 80, 79, 93, 95, 106, 111, 125, 127, 145, 149, 166, 172, 191, 196, 222, 229, 250, 262, 289, 298, 330, 343, 373, 391, 427, 442, 486

Offset

0,4

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.

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 23, 25.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (corrected and extended previous b-file 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.

Formula

G.f.: chi_0(q) = Sum_{n>=0} q^n/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))).

G.f.: chi_0(q) = 1 + Sum_{n>=0} q^(2n+1)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).

a(n) is the number of partitions of 5n with rank == 1 (mod 5) minus number with rank == 0 (mod 5).

a(n) is the number of partitions of n with unique smallest part and all other parts <= twice the smallest part.

a(n) is the number of partitions where the largest part is odd and all other parts are greater than half of the largest part. - N. Sato, Jan 21 2010

a(n) ~ exp(Pi*sqrt(2*n/15)) / sqrt((5 + sqrt(5))*n). - Vaclav Kotesovec, Jun 12 2019

Mathematica

1+Series[Sum[q^(2n+1)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)/Product[1-x^j, {j, k+1, 2*k+1}], {k, 0, Floor[nmax/2]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053263, A053264, A053265, A053266, A053267.

Sequence in context: A319318 A029164 A364350 * A007359 A213424 A174427

Adjacent sequences: A053259 A053260 A053261 * A053263 A053264 A053265

Keyword

nonn,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved