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A053276 Coefficients of the '7th-order' mock theta function F_1(q). 6
0, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 2, 4, 4, 4, 4, 6, 5, 6, 6, 7, 8, 9, 8, 10, 11, 11, 12, 14, 13, 16, 16, 18, 19, 21, 20, 24, 25, 26, 28, 31, 31, 35, 36, 39, 41, 45, 45, 50, 53, 55, 58, 64, 65, 71, 73, 79, 83, 89, 90, 99, 103, 109, 114, 123, 126, 135, 141, 149, 157, 167, 171, 185 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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 G. C. Greubel)

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_1(q) = Sum_{n >= 1} q^n^2/((1-q^n)(1-q^(n+1))...(1-q^(2n-1))).

a(n) = number of partitions of 7n-1 with rank == 2 (mod 7) minus number with rank == 3 (mod 7).

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

MATHEMATICA

Series[Sum[q^n^2/Product[1-q^k, {k, n, 2n-1}], {n, 1, 10}], {q, 0, 100}]

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

CROSSREFS

Other '7th-order' mock theta functions are at A053275, A053277, A053278, A053279, A053280.

Sequence in context: A068796 A154804 A207642 * A064065 A284486 A232194

Adjacent sequences:  A053273 A053274 A053275 * A053277 A053278 A053279

KEYWORD

nonn,easy

AUTHOR

Dean Hickerson, Dec 19 1999

STATUS

approved

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Last modified June 12 21:19 EDT 2021. Contains 344967 sequences. (Running on oeis4.)