A183011 (24n - 1)p(n): traces of partition class polynomials, with a(0) = -1.
-1, 23, 94, 213, 475, 833, 1573, 2505, 4202, 6450, 10038, 14728, 22099, 31411, 45225, 63184, 88473, 120879, 165935, 222950, 300333, 398376, 528054, 691505, 905625, 1172842, 1517628, 1947470, 2494778, 3172675, 4029276, 5083606, 6403683, 8023113
Offset: 0
Keywords
Examples
1. For n = 6, the number of partitions of 6 is 11, so a(6) = (24*6 - 1)*11 = 143*11 = 1573. 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. G.f. = -1 + 23*x + 94*x^2 + 213*x^3 + 475*x^4 + 833*x^5 + 1573*x^6 + 2505*x^7 + ... G.f. = -q^-1 + 23*q^23 + 94*q^47 + 213*q^71 + 475*q^95 + 833*q^119 + 1573*q^143 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..2500
- K. Bringmann and K. Ono, An arithmetic formula for the partition function, Proc Am. Math. Soc. 135 (2007), 3507-3515
- K. Bringmann and K. Ono, Lifting elliptic cusp forms to Maass forms with an application to partitions, Proc Nat. Acad. Sci. 104 (10) (2007) 3725-3731
- J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
- J. H. Bruinier, K. Ono, A. V. Sutherland, Class polynomials for nonholomorphic modular functions, arXiv:1301.5672 [math.NT], 2013-2015.
- A. Dabholkar, S. Murthy, D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074 [hep-th], 2012-2014, p. 46.
- A. Folsom, Z. A. Kent and K. Ono, l-adic properties of the partition function, preprint.
- A. Folsom, Z. A. Kent and K. Ono, l-adic properties of the partition function, Advances in Mathematics, 229 (2012), pages 1586-1609.
- F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2010; see (1.5), (2.10).
- P. M. Jenkins, Traces of singular moduli, modular forms, and Maass forms
- E. Larson and L. Rolen, Integrality properties of the CM-values of certain weak Maass forms, arXiv:1107.4114 [math.NT], 2011.
- K. Ono, Congruences for the Andrews spt-function, (see 2.1 Producing modular forms)
- A. V. Sutherland, Partition class polynomials, Hpart_n(x), n = 1..770
Crossrefs
Programs
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Mathematica
a[ n_] := (24 n - 1) SeriesCoefficient[ 1 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jun 26 2017 *) -
PARI
{a(n) = if( n<0, 0, (24*n - 1) * numbpart(n))}; /* Michael Somos, Aug 28 2013 */
Formula
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
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
Comments