A006306 Coefficients of the '2nd-order' mock theta function mu(q). (Formerly M0163)

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

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: A282738 A093890 A325609 * A322100 A277100 A337363

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