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 A183011 (24n - 1)p(n): traces of partition class polynomials, with a(0) = -1. 23
 -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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. 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 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 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 A. Dabholkar, S. Murthy, D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, p. 46. A. Folsom, Z. A. Kent and K. Ono, l-adic properties of the partition function F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, 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 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 FORMULA a(n) = A183010(n)*A000041(n). a(n) = 24*A066186(n) - A000041(n) = A183009(n) - A000041(n) = (A008606(n)-1)*A000041(n). 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 EXAMPLE 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 + ... MATHEMATICA a[ n_] := (24 n - 1) SeriesCoefficient[ 1 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jun 26 2017 *) PROG (PARI) {a(n) = if( n<0, 0, (24*n - 1) * numbpart(n))}; /* Michael Somos, Aug 28 2013 */ CROSSREFS Positive terms are the partial sums of A183012, also the column 24 of A182729. Cf. A000041, A008606, A066186, A183006, A183009, A183010, A183054, A187206. Sequence in context: A042030 A042032 A257976 * A158544 A154376 A155815 Adjacent sequences:  A183008 A183009 A183010 * A183012 A183013 A183014 KEYWORD sign AUTHOR Omar E. Pol, Jan 21 2011 STATUS approved

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Last modified December 12 18:43 EST 2018. Contains 318081 sequences. (Running on oeis4.)