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Coefficients of the '2nd-order' mock theta function mu(q).
(Formerly M0163)
4

%I M0163 #26 Dec 18 2021 22:17:03

%S 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,

%T -32,17,38,-18,-50,20,58,-25,-63,33,77,-35,-94,36,108,-48,-122,60,141,

%U -63,-170,70,195,-87,-215,101,250,-110,-294,124,333,-146,-371,173,424,-190,-492,206,554,-245,-617,283

%N Coefficients of the '2nd-order' mock theta function mu(q).

%C Contribution from _Jeremy Lovejoy_, Dec 19 2008: (Start)

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

%C |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)

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

%H T. D. Noe, <a href="/A006306/b006306.txt">Table of n, a(n) for n = 0..1000</a>

%H G. E. Andrews, <a href="http://dx.doi.org/10.1007/BFb0096452">Mordell integrals and Ramanujan's "Lost" Notebook</a>, pp. 10-48 of Analytic Number Theory (Philadelphia 1980), Lect. Notes Math. 899 (1981).

%H K. Bringmann, K. Ono and R. Rhoades, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/108.pdf">Eulerian series as modular forms</a>, J. Amer. Math. Soc. 21 (2008), 1085-1104. [From _Jeremy Lovejoy_, Dec 19 2008]

%H J. Lovejoy and R. Osburn, <a href="http://arxiv.org/abs/0705.4535">M_2-rank differences for partitions without repeated odd parts</a> [From _Jeremy Lovejoy_, Dec 19 2008]

%H R. J. McIntosh, <a href="http://dx.doi.org/10.4153/CMB-2007-028-9">Second order mock theta functions</a>, Canad. Math. Bull. 50 (2007), 284-290. [From _Jeremy Lovejoy_, Dec 19 2008]

%F 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).

%e 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 + ...

%t 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]

%t 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 *)

%Y Cf. A006304, A006305.

%K sign,easy,nice

%O 0,4

%A _N. J. A. Sloane_

%E Corrected and extended by _Dean Hickerson_, Dec 13 1999