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

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A006306 Coefficients of the '2nd order' mock theta function mu(q).
(Formerly M0163)
4
1, -1, 1, 2, -1, -4, 1, 5, -2, -5, 4, 7, -4, -11, 3, 13, -6, -14, 9, 18, -7, -24, 8, 29, -14, -32, 17, 38, -18, -50, 20, 58, -25, -63, 33, 77, -35, -94, 36, 108, -48, -122, 60, 141, -63, -170, 70, 195, -87, -215, 101, 250, -110, -294, 124, 333, -146, -371, 173, 424, -190, -492, 206, 554, -245, -617, 283 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Contribution from Jeremy Lovejoy, Dec 19 2008: (Start)

Coefficients of the "second order" mock theta function mu(q).

|a(n)| is the number of partitions of n without repeated odd parts whose M2-rank is even minus the number of partitions of n without repeated odd parts whose M2-rank is odd. (End)

REFERENCES

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

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

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

K. Bringmann, K. Ono and R. Rhoades, Eulerian series as modular forms, J. Amer. Math. Soc. 21 (2008), 1085-1104. [From Jeremy Lovejoy, Dec 19 2008]

J. Lovejoy and R. Osburn, M_2-rank differences for partitions without repeated odd parts [From Jeremy Lovejoy, Dec 19 2008]

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

FORMULA

G.f.: Sum_{n >= 0} (-1)^n q^n^2 (1-q)(1-q^3)...(1-q^(2n-1))/((1+q^2)^2 (1+q^4)^2 ... (1+q^(2n))^2).

EXAMPLE

G.f. = 1 - x + x^2 + 2*x^3 - x^4 - 4*x^5 + x^6 + x*x^7 - 2*x^8 - 5*x^9 + ...

MATHEMATICA

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

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^k^2 QPochhammer[ x, x^2, k] / QPochhammer[- x^2, x^2, k]^2, {k, 0, Sqrt[ n]}], {x, 0, n}]]; (* Michael Somos, Jul 09 2015 *)

CROSSREFS

Cf. A006304, A006305.

Sequence in context: A261242 A088296 A093890 * A277100 A214579 A083711

Adjacent sequences:  A006303 A006304 A006305 * A006307 A006308 A006309

KEYWORD

sign,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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 3 01:12 EST 2016. Contains 278694 sequences.