A183007 -id:A183007 - cp's OEIS frontend

A183010 a(n) = 24*n - 1.

-1, 23, 47, 71, 95, 119, 143, 167, 191, 215, 239, 263, 287, 311, 335, 359, 383, 407, 431, 455, 479, 503, 527, 551, 575, 599, 623, 647, 671, 695, 719, 743, 767, 791, 815, 839, 863, 887, 911, 935, 959, 983, 1007, 1031, 1055, 1079, 1103, 1127, 1151, 1175, 1199

Offset: 0

Omar E. Pol, Jan 21 2011

a(n) is also the denominator 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 numerators see A183011.
It appears that a(n) is also the denominator of the coefficient of the third term in the n-th Bruinier-Ono "partition polynomial" H_n(x). See the Bruinier-Ono paper, chapter 5 "Examples". For the numerators see A183007. - Omar E. Pol, Jul 13 2011
Also exponents in the formula q^(-1) + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 11*q^143 + 15*q^167 + ... in which the coefficients are the partition numbers (see A000041, Example section). - Omar E. Pol, Feb 27 2013

			G.f. = -1 + 23*x + 47*x^2 + 71*x^3 + 95*x^4 + 119*x^5 + 143*x^6 + 167*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
A. Dabholkar, S. Murthy, and D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074 [hep-th], 2012-2014, see p. 46.
H. Gupta, Congruent properties of sigma(n), Math. Student 13 (1945) 25-29.
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)
W. Sierpinski, Elementary Theory of numbers, Monografie Mathematyczne, vol. 42 (1964) chapt 4, p. 168.
Leo Tavares, Illustration: Star Pairs
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000041, A000203, A008606, A134517 (subset of primes), A183009, A183011, A187206, A280097 (sum of divisors), A280098.

Cf. A008594.

[24*n-1: n in [0..50]]; // G. C. Greubel, Aug 14 2018

Range[23, 2000, 24] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)
(24*Range[0,50])-1 (* Harvey P. Dale, Mar 28 2015 *)

a(n)=24*n-1 \\ Charles R Greathouse IV, Jun 14 2011

a(n) = A008606(n) - 1.
a(1)=23, a(2)=47, a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jan 23 2011
a(n) = A183011(n)/A000041(n). - Omar E. Pol, Jul 14 2011
24 * A280098(n) = A000203(a(n)) if n>0. - Michael Somos, Dec 25 2016
E.g.f.: (24*x-1)*exp(x). - G. C. Greubel, Aug 14 2018
G.f.: (-1 + 25*x)/(1-x)^2. - Wolfdieter Lang, Dec 10 2021
a(n) = 2*A008594(n) - 1. - Leo Tavares, Jun 06 2023

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.

A222031 Irregular triangle read by rows in which row n gives numerators of the coefficients of the partition class polynomial Hpart_n(x), n >= 1.

Original entry on oeis.org

1, -23, 3592, -419, 1, -94, 169659, -65838, 1092873176, 145023, 1, -213, 1312544, -723721, 44648582886, 9188934683, 166629520876208, 2791651635293, 1, -475, 9032603, -9455070, 3949512899743, -97215753021, 9776785708507683, -53144327916296, -134884469547631

Offset: 1

Omar E. Pol, Mar 04 2013

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.

Note that the absolute value of T(n,2) is also the trace Tr(n) = A183011(n), the numerator of the finite algebraic formula for the number of partitions of n. The formula is p(n) = Tr(n)/(24*n - 1). See theorem 1.1 in the Bruinier-Ono paper.

			For n = 1 the first partition class polynomial Hpart_1(x) is x^3 - 23*x^2 + 3592/23*x - 419, so the numerators of the coefficients are 1, -23, 3592, -419.
Triangle begins:
1, -23, 3592, -419;
1, -94, 169659, -65838, 1092873176, 145023;
1, -213, 1312544, -723721, 44648582886, 9188934683, 166629520876208, 2791651635293;
1, -475, 9032603, -9455070, 3949512899743, -97215753021, 9776785708507683, -53144327916296, -134884469547631;
...

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), for n = 1..770

Row n has length 1 + A188569(n). Absolute values of column 2 give A183011. Columns 3-4: A183007, A187218. For denominators see A222032.

Cf. A000041, A183010, A220515.

abs(T(n,2))/(24n-1) = A183011(n)/A183010(n) = A000041(n).

A222032 Irregular triangle read by rows in which row n gives denominators of the coefficients of the partition class polynomial Hpart_n(x), n >= 1.

Original entry on oeis.org

1, 1, 23, 1, 1, 1, 47, 1, 2209, 47, 1, 1, 71, 1, 5041, 71, 357911, 5041, 1, 1, 95, 1, 9025, 19, 857375, 361, 11875, 1, 1, 119, 1, 2023, 1, 240737, 14161, 200533921, 1685159, 4857223, 1, 1, 143, 1, 1573, 143, 2924207, 20449, 418161601, 2924207, 27217619

Offset: 1

Omar E. Pol, Mar 04 2013

For more information see A222031.

			For n = 1 the first partition class polynomial Hpart_1(x) is x^3 - 23*x^2 + 3592/23*x - 419, so the denominators of the coefficients are 1, 1, 23, 1.
Triangle begins:
1, 1, 23, 1;
1, 1, 47, 1, 2209, 47;
1, 1, 71, 1, 5041, 71, 357911, 5041;
1, 1, 95, 1, 9025, 19, 857375, 361, 11875;
1, 1, 119, 1, 2023, 1, 240737, 14161, 200533921, 1685159, 4857223;
1, 1, 143, 1, 1573, 143, 2924207, 20449, 418161601, 2924207, 27217619;
1, 1, 167, 1, 27889, 167, 4657463, 27889, 777796321, 4657463, 129891985607, 777796321;

A. V. Sutherland, Partition class polynomials, Hpart_n(x), for n = 1..770

Row n has length 1 + A188569(n). For numerators see A222031.

Cf. A000041, A183007, A183010, A183011, A187218, A188569, A220515.

cp's OEIS Frontend

A183010 a(n) = 24*n - 1.

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

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A222031 Irregular triangle read by rows in which row n gives numerators of the coefficients of the partition class polynomial Hpart_n(x), n >= 1.

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A222032 Irregular triangle read by rows in which row n gives denominators of the coefficients of the partition class polynomial Hpart_n(x), n >= 1.

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