login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A006304 Coefficients of the '2nd order' mock theta function A(q).
(Formerly M0685)
4
0, 1, 2, 3, 5, 8, 11, 16, 23, 31, 43, 58, 76, 101, 132, 170, 219, 280, 354, 447, 562, 699, 869, 1076, 1323, 1625, 1987, 2418, 2937, 3556, 4289, 5162, 6196, 7413, 8853, 10547, 12530, 14860, 17586, 20763, 24474, 28792, 33802, 39624, 46368, 54163 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The "second order" mock theta function A(q). [From Jeremy Lovejoy, Dec 19 2008]

REFERENCES

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 8.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

G. E. Andrews, Mordell integrals and Ramanujan's "Lost" Notebook, pp. 10-48 of Analytic Number Theory (Philadelphia 1980), Lect. Notes Math. 899 (1981).

R. J. McIntosh, Second order mock theta functions, Canad. Math. Bull. 50 (2007), 284-290. [From Jeremy Lovejoy, Dec 19 2008]

Wikipedia, Mock modular form

FORMULA

G.f.: sum for n >= 0 of q^(n+1) (1+q^2)(1+q^4)...(1+q^(2n))/((1-q)(1-q^3)...(1-q^(2n+1)))

G.f.: sum for n >= 0 of q^(n+1)^2 (1+q)(1+q^3)...(1+q^(2n-1))/((1-q)(1-q^3)...(1-q^(2n+1)))^2

a(n) ~ exp(Pi*sqrt(n/2)) / (8*sqrt(n)). - Vaclav Kotesovec, Jun 11 2019

EXAMPLE

G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 11*x^6 + 16*x^7 + 23*x^8 + ...

MATHEMATICA

Series[Sum[q^(n+1)^2 Product[1+q^(2k-1), {k, 1, n}]/Product[1-q^(2k-1), {k, 1, n+1}]^2, {n, 0, 9}], {q, 0, 100}]

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k + 1)^2 QPochhammer[ -x, x^2, k] / QPochhammer[ x, x^2, k + 1]^2, {k, 0, Sqrt[ n] - 1}], {x, 0, n}]]; (* Michael Somos, Apr 08 2015 *)

nmax = 100; CoefficientList[Series[Sum[x^(k+1)^2 * Product[1 + x^(2*j - 1), {j, 1, k}] / Product[1 - x^(2*j - 1), {j, 1, k+1}]^2, {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 11 2019 *)

CROSSREFS

Cf. A006305, A006306.

Sequence in context: A175831 A070228 A173599 * A238591 A039847 A046938

Adjacent sequences:  A006301 A006302 A006303 * A006305 A006306 A006307

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Corrected and extended by Dean Hickerson, Dec 13 1999

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 19 04:52 EDT 2019. Contains 327187 sequences. (Running on oeis4.)