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%I #91 May 13 2019 06:21:13
%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 G. C. Greubel, <a href="/A183011/b183011.txt">Table of n, a(n) for n = 0..2500</a>
%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.mathcs.emory.edu/~ono/publications-cv/pdfs/134.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>, arXiv:1301.5672 [math.NT], 2013-2015.
%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>, arXiv:1208.4074 [hep-th], 2012-2014, 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>, preprint.
%H A. Folsom, Z. A. Kent and K. Ono, <a href="https://doi.org/10.1016/j.aim.2011.11.013">l-adic properties of the partition function</a>, Advances in Mathematics, 229 (2012), pages 1586-1609.
%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>, arXiv:1011.1957 [math.NT], 2010; 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>, arXiv:1107.4114 [math.NT], 2011.
%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
%F G.f.: Sum_{k >= 0} a(k) * q^(24*k - 1) = q * d/dq (1/q * Product_{k > 0} 1 / (1 - q^(24*k))). - _Michael Somos_, Aug 28 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.
%e G.f. = -1 + 23*x + 94*x^2 + 213*x^3 + 475*x^4 + 833*x^5 + 1573*x^6 + 2505*x^7 + ...
%e G.f. = -q^-1 + 23*q^23 + 94*q^47 + 213*q^71 + 475*q^95 + 833*q^119 + 1573*q^143 + ...
%t a[ n_] := (24 n - 1) SeriesCoefficient[ 1 / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Jun 26 2017 *)
%o (PARI) {a(n) = if( n<0, 0, (24*n - 1) * numbpart(n))}; /* _Michael Somos_, Aug 28 2013 */
%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
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