A183054 -id:A183054 - cp's OEIS frontend

A220515 Numbers n such that A183054(n) is not equal to A188569(n).

24, 47, 49, 74, 96, 99, 116, 124, 145, 149, 162, 174, 194, 199, 224, 237, 243, 249, 274, 277, 292, 299, 324, 331, 341, 346, 349, 358, 374, 390, 399, 424, 439, 449, 474, 479, 488, 499, 500, 507, 524, 537, 549, 566, 574, 586, 599, 600, 624, 635

Offset: 1

Omar E. Pol, Feb 27 2013

nonn

For an algorithm to compute the partition class polynomial Hpart_n(x) see the Bruinier-Ono-Sutherland paper, 3.3. Algorithm 3, p. 15-19. For more information see A222031.

			First three terms are 24, 47, 49 because first 50 terms of A183054 coincide with first 50 terms of A188569 except for the indices 24, 47, 49 as shown below:
(A183054(24) = 3) < (A188569(24) = 21).
(A183054(47) = 3) < (A188569(47) = 27).
(A183054(49) = 5) < (A188569(49) = 35).
Observation:
A183054(24) = A188569(24)/7 = 3.
A183054(47) = A188569(47)/9 = 3.
A183054(49) = A188569(49)/7 = 5.

J. H. Bruinier, K. Ono, A. V. Sutherland, Class polynomials for nonholomorphic modular functions
A. V. Sutherland, Partition class polynomials, Hpart_n(x), for n = 1..770

Cf. A000041, A183010, A183011, A183054, A188569, A222031, A222032.

a(4)-a(50) from Giovanni Resta, Mar 04 2013

A183011 (24n - 1)p(n): traces of partition class polynomials, with a(0) = -1.

Original entry on oeis.org

-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

Omar E. Pol, Jan 21 2011

sign

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

			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 + ...

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

Positive terms are the partial sums of A183012, also the column 24 of A182729.

Cf. A000041, A008606, A066186, A183006, A183009, A183010, A183054, A187206.

a[ n_] := (24 n - 1) SeriesCoefficient[ 1 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jun 26 2017 *)

{a(n) = if( n<0, 0, (24*n - 1) * numbpart(n))}; /* Michael Somos, Aug 28 2013 */

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

A188569 Degree of the n-th partition class polynomial Hpart_n(x).

Original entry on oeis.org

3, 5, 7, 8, 10, 10, 11, 13, 14, 15, 13, 14, 19, 18, 19, 17, 16, 21, 20, 25, 21, 18, 26, 21, 25, 22, 23, 30, 24, 31, 21, 22, 32, 30, 33, 21, 29, 31, 28, 36, 27, 30, 35, 36, 34, 23, 27, 41, 35, 38, 35, 26, 40, 36, 45, 34, 25, 44, 34, 39, 32, 37, 49, 38, 51, 33

Offset: 1

Omar E. Pol, Feb 21 2013

nonn

a(n) is the degree of the n-th partition class polynomial whose trace is the numerator of the finite algebraic formula for the number of partitions of n. The formula for the partition function is p(n) = Tr(n)/(24n - 1). See theorem 1.1 in the Bruinier-Ono paper. The traces are in A183011. See also Sutherland's table of Hpart_n(x) in the Links section.

First differs from A183054 at a(24). It appears that this coincides with A183054 in a large number of terms.

			In the Bruinier-Ono paper, chapter 5 "Examples", the first "partition polynomial" is H_1(x) = x^3 - 23*x^2 + (3592/23)*x - 419, which has degree 3, so a(1) = 3.

Giovanni Resta, Table of n, a(n) for n = 1..750. Data from A. V. Sutherland's website
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. V. Sutherland, Partition class polynomials, Hpart_n(x), n = 1..770

Cf. A183007, A183010, A183011, A183054, A187218.

This sequence arises from the original definition of A183054 (Jul 14 2011) which was changed.

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A220515 Numbers n such that A183054(n) is not equal to A188569(n).

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A183011 (24n - 1)p(n): traces of partition class polynomials, with a(0) = -1.

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A188569 Degree of the n-th partition class polynomial Hpart_n(x).

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