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A053263 Coefficients of the '5th order' mock theta function chi_1(q). 15
1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 8, 9, 9, 12, 12, 15, 15, 18, 19, 23, 23, 27, 30, 33, 34, 41, 42, 49, 51, 57, 61, 69, 72, 81, 87, 96, 100, 113, 119, 132, 140, 153, 163, 180, 188, 208, 221, 240, 253, 278, 294, 319, 339, 366, 388, 422, 443, 481, 510, 549, 580, 626, 662 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The rank of a partition is its largest part minus the number of parts.

Number of partitions of n such that 2*(least part) > greatest part. - Clark Kimberling, Feb 16 2014

REFERENCES

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

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 25

George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304

LINKS

Table of n, a(n) for n=0..62.

FORMULA

G.f.: chi_1(q) = sum for n >= 0 of q^n/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).

G.f.: chi_1(q) = 1 + sum for n >= 0 of q^(2n+1) (1+q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))).

a(n) = twice the number of partitions of 5n+3 with rank == 2 (mod 5) minus number with rank == 0 or 1 (mod 5).

a(n) - 1 = number of partitions of n with unique smallest part and all other parts <= one plus twice the smallest part.

MATHEMATICA

1+Series[Sum[q^(2n+1)(1+q^n)/Product[1-q^k, {k, n+1, 2n+1}], {n, 0, 49}], {q, 0, 100}]

(* Also: *)

Table[Count[ IntegerPartitions[n], p_ /; 2 Min[p] > Max[p]], {n, 40}]

(* Clark Kimberling, Feb 16 2014 *)

CROSSREFS

Other '5th order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053264, A053265, A053266, A053267.

Sequence in context: A083802 A198318 A100881 * A056039 A181972 A058747

Adjacent sequences:  A053260 A053261 A053262 * A053264 A053265 A053266

KEYWORD

nonn,easy

AUTHOR

Dean Hickerson, Dec 19 1999

STATUS

approved

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Last modified November 26 09:02 EST 2014. Contains 250021 sequences.