

A053262


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


15



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
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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. 354355
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 23, 25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 bfile from G. C. Greubel)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242255.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274304.


FORMULA

G.f.: chi_0(q) = sum for n >= 0 of q^n/((1q^(n+1))(1q^(n+2))...(1q^(2n)))
G.f.: chi_0(q) = 1 + sum for n >= 0 of q^(2n+1)/((1q^(n+1))(1q^(n+2))...(1q^(2n+1)))
a(n) = number of partitions of 5n with rank == 1 (mod 5) minus number with rank == 0 (mod 5)
a(n) = number of partitions of n with unique smallest part and all other parts <= twice the smallest part
a(n) = 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[1q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)/Product[1x^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: A244327 A319318 A029164 * A007359 A213424 A174427
Adjacent sequences: A053259 A053260 A053261 * A053263 A053264 A053265


KEYWORD

nonn,easy


AUTHOR

Dean Hickerson, Dec 19 1999


STATUS

approved



