|
%I
%S -1,23,94,213,475,833,1573,2505,4202,6450,10038,14728,22099,31411,
%T 45225,63184,88473,120879,165935,222950,300333,398376,528054,691505,
%U 905625,1172842,1517628,1947470,2494778,3172675,4029276,5083606,6403683,8023113
%N (24n - 1)p(n): traces of partition class polynomials, with a(0) = -1.
%C a(n) is also Tr(n), the numerator of the finite algebraic formula for the number of partitions of n, if n >= 1. The formula is p(n) = Tr(n)/(24*n - 1), n >= 1. See theorem 1.1 of the Bruinier-Ono paper in the link. For the denominators see A183010.
%C a(n) is also the coefficient of the second term (the trace) in the n-th Bruinier-Ono "partition polynomial" H_n(x), if n >= 1. See the Bruinier-Ono paper, theorem 1.1 and chapter 5 "Examples". For the coefficients of the 4th terms see A187218. [Omar E. Pol, Jul 10 2011]
%C In the Bruinier-Ono-Sutherland paper (Jan 23 2013) partition polynomials are called "partition class polynomials". See also Sutherland's table of Hpart_n(x) in link section. - _Omar E. Pol_, Feb 20 2013
%H K. Bringmann and K. Ono, <a href="http://www.math.wisc.edu/~ono/reprints/097.pdf">An arithmetic formula for the partition function</a>, Proc Am. Math. Soc. 135 (2007), <a href="http://dx.doi.org/10.1090/S0002-9939-07-08883-1">3507-3515</a>
%H K. Bringmann and K. Ono, <a href="http://www.math.wisc.edu/~ono/reprints/106.pdf">Lifting elliptic cusp forms to Maass forms with an application to partitions</a>, Proc Nat. Acad. Sci. 104 (10) (2007) <a href="http://dx.doi.org/10.1073/pnas.0611414104">3725-3731</a>
%H J. H. Bruinier and K. Ono, <a href="http://www.aimath.org/news/partition/brunier-ono.pdf">Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms</a>
%H J. H. Bruinier, K. Ono, A. V. Sutherland, <a href="http://arxiv.org/abs/1301.5672">Class polynomials for nonholomorphic modular functions</a>
%H A. Dabholkar, S. Murthy, D. Zagier, <a href="http://arxiv.org/abs/1208.4074">Quantum Black Holes, Wall Crossing, and Mock Modular Forms</a>, p. 46.
%H A. Folsom, Z. A. Kent and K. Ono, <a href="http://www.aimath.org/news/partition/folsom-kent-ono.pdf">l-adic properties of the partition function</a>
%H F. G. Garvan, <a href="http://arxiv.org/abs/1011.1957">Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences</a>, see (1.5), (2.10).
%H P. M. Jenkins, <a href="http://math.byu.edu/~jenkins/JenkinsPhDThesis.pdf">Traces of singular moduli, modular forms, and Maass forms</a>
%H E. Larson and L. Rolen, <a href="http://arxiv.org/abs/1107.4114">Integrality properties of the CM-values of certain weak Maass forms</a>
%H K. Ono, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/131.pdf">Congruences for the Andrews spt-function</a>, (see 2.1 Producing modular forms)
%H A. V. Sutherland, <a href="http://math.mit.edu/~drew/Pfiles">Partition class polynomials</a>, Hpart_n(x), n = 1..770
%F a(n) = A183010(n)*A000041(n).
%F a(n) = 24*A066186(n)-A000041(n) = A183009(n)-A000041(n) = (A008606(n)-1)*A000041(n).
%F a(n) = 12M_2(n) - p(n) = 24spt(n) + 12N_2(n) - p(n) = 12*A220909(n) - A000041(n) = 24*A092269(n) + 12*A220908(n) - A000041(n), n >= 1. - _Omar E. Pol_, Feb 17 2013
%e 1) For n = 6, the number of partitions of 6 is 11, so a(6) = (24*6 - 1)*11 = 143*11 = 1573.
%e 2) For n = 1, in the Bruinier-Ono paper, chapter 5, the first "partition polynomial" is H_1(x) = x^3 - 23*x^2 + (3592/23)*x - 419. The coefficient of the second term (the trace) is 23, so a(1) = 23.
%Y Positive terms are the partial sums of A183012, also the column 24 of A182729.
%Y Cf. A000041, A008606, A066186, A183006, A183009, A183010, A183054, A187206.
%K sign
%O 0,2
%A _Omar E. Pol_, Jan 21 2011
|
