A183009 -id:A183009 - 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

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

A182728 Array T(n,k) = nkA000041(n) read by antidiagonals, n,k >= 1.

Original entry on oeis.org

1, 4, 2, 9, 8, 3, 20, 18, 12, 4, 35, 40, 27, 16, 5, 66, 70, 60, 36, 20, 6, 105, 132, 105, 80, 45, 24, 7, 176, 210, 198, 140, 100, 54, 28, 8, 270, 352, 315, 264, 175, 120, 63, 32, 9, 420, 540, 528, 420, 330, 210, 140, 72, 36, 10, 616, 840, 810, 704, 525, 396, 245, 160, 81, 40, 11

Offset: 1

Omar E. Pol, Jan 22 2011

			Array begins:
    1,   2,   3,   4,   5,   6,   7, ...
    4,   8,  12,  16,  20,  24,  28, ...
    9,  18,  27,  36,  45,  54,  63, ...
   20,  40,  60,  80, 100, 120, 140, ...
   35,  70, 105, 140, 175, 210, 245, ...
   66, 132, 198, 264, 330, 396, 462, ...
  105, 210, 315, 420, 525, 630, 735, ...

Cf. A000041, A135010, A182729, A183011.

Column 1 gives A066186. Column 24 gives A183009.

A182728 := proc(n,k) n*k*combinat[numbpart](n) ; end proc:
seq(seq( A182728(d+1-k, k), k=1..d), d=1..10) ; # R. J. Mathar, Feb 01 2011

A182729 Square array T(n,k) = (nk-1)A000041(n) read by antidiagonals upwards.

Original entry on oeis.org

0, 2, 1, 6, 6, 2, 15, 15, 10, 3, 28, 35, 24, 14, 4, 55, 63, 55, 33, 18, 5, 90, 121, 98, 75, 42, 22, 6, 154, 195, 187, 133, 95, 51, 26, 7, 240, 330, 300, 253, 168, 115, 60, 30, 8, 378, 510, 506, 405, 319, 203, 135, 69, 34, 9

Offset: 1

Omar E. Pol, Jan 22 2011

			Square array T(n,k) begins:
   0,   1,   2,   3,   4,   5, ...
   2,   6,  10,  14,  18,  22, ...
   6,  15,  24,  33,  42,  51, ...
  15,  35,  55,  75,  95, 115, ...
  28,  63,  98, 133, 168, 203, ...
  55, 121, 187, 253, 319, 385, ...

Cf. A000041, A135010, A182724, A182728, A183009, A183010, A183011.

T:= (n,k)-> (n*k-1)*combinat[numbpart](n):
seq (seq (T(d-k, k), k=1..d-1), d=2..11);

Table[With[{n = m - k + 1}, (n k - 1) PartitionsP[n]], {m, 10}, {k, m}] // Flatten (* Michael De Vlieger, Nov 02 2017 *)

T(n,1) = A182724(n).

T(n,24) = A183011(n).

A187206 a(n) = 6(24n - 1).

Original entry on oeis.org

138, 282, 426, 570, 714, 858, 1002, 1146, 1290, 1434, 1578, 1722, 1866, 2010, 2154, 2298, 2442, 2586, 2730, 2874, 3018, 3162, 3306, 3450, 3594, 3738, 3882, 4026, 4170, 4314, 4458, 4602, 4746, 4890, 5034, 5178, 5322, 5466, 5610, 5754, 5898, 6042, 6186, 6330, 6474, 6618

Offset: 1

Omar E. Pol, Jul 09 2011

The expression 6*(24*n - 1) is mentioned in the Bruinier-Ono paper (see theorem 1.1 and chapter 5).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms.
E. Larson and L. Rolen, Integrality properties of the CM-values of certain weak Maass forms.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000041, A183009, A183010, A183011.

[6*(24*n - 1): n in [1..45]]; // Vincenzo Librandi, Jul 12 2011

144*Range[40]-6 (* Harvey P. Dale, Jul 20 2011 *)

a(n)=144*n-6 \\ Charles R Greathouse IV, Nov 03 2011

a(n) = 6*A183010(n).
From Elmo R. Oliveira, Apr 03 2025: (Start)
G.f.: 6*x*(x + 23)/(1 - x)^2.
E.g.f.: 6*(exp(x)*(24*x - 1) + 1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

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

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

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A182728 Array T(n,k) = n*k*A000041(n) read by antidiagonals, n,k >= 1.

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A182729 Square array T(n,k) = (n*k-1)*A000041(n) read by antidiagonals upwards.

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A187206 a(n) = 6*(24*n - 1).

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A183006 a(n) = 24*A138879(n).

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A182728 Array T(n,k) = nkA000041(n) read by antidiagonals, n,k >= 1.

A182729 Square array T(n,k) = (nk-1)A000041(n) read by antidiagonals upwards.

A187206 a(n) = 6(24n - 1).