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A053263 Coefficients of the '5th order' mock theta function chi_1(q). 16
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 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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 20, 25

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Seiichi Manyama)

George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.

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

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.

a(n) ~ sqrt(phi/2) * exp(Pi*sqrt(2*n/15)) / (5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 16 2019

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 *)

nmax = 100; CoefficientList[Series[1 + Sum[x^(2*k+1)*(1+x^k) / 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, A053262, A053264, A053265, A053266, A053267.

Sequence in context: A326668 A198318 A100881 * A317908 A056039 A181972

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 July 9 14:56 EDT 2020. Contains 335543 sequences. (Running on oeis4.)