A182729 -id:A182729 - cp's OEIS frontend

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

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

A182727 Sum of largest parts of the shell model of partitions with n regions.

Original entry on oeis.org

1, 3, 6, 8, 12, 15, 20, 22, 26, 29, 35, 38, 43, 47, 54, 56, 60, 63, 69, 74, 78, 86, 89, 94, 98, 105, 108, 114, 119, 128, 130, 134, 137, 143, 148, 152, 160, 164, 171, 177, 182, 192, 195, 200, 204, 211, 214, 220, 225, 234, 239, 243, 251, 258, 264, 275, 277, 281

Offset: 1

Omar E. Pol, Jan 25 2011

Question: Is there some connection with fractals?

			For n = 6 the largest parts of the first six regions of the shell model of partitions are 1, 2, 3, 2, 4, 3, so a(6) = 1+2+3+2+4+3 = 15.
Written as a triangle begins:
1;
3;
6;
8,   12;
15,  20;
22,  26, 29, 35;
38,  43, 47, 54;
56,  60, 63, 69, 74, 78, 86;
89,  94, 98,105,108,114,119,128;
130,134,137,143,148,152,160,164,171,177,182,192;
195,200,204,211,214,220,225,234,239,243,251,258,264,275;

Alois P. Heinz, Table of n, a(n) for n = 1..4000
Omar E. Pol, Illustration of the seven regions of 5

Partial sums of A141285. Row j has length A187219(j). Right border gives A006128.

Cf. A135010, A141285, A182728, A182729, A186114, A206437, A182181, A182244, A210990.

a(A000041(n)) = A182181(A000041(n)) = A006128(n). - Omar E. Pol, May 24 2012

New name from Omar E. Pol, Apr 26 2012

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

A182724 Sum of all parts of all partitions of n minus the number of partitions of n.

Original entry on oeis.org

0, 2, 6, 15, 28, 55, 90, 154, 240, 378, 560, 847, 1212, 1755, 2464, 3465, 4752, 6545, 8820, 11913, 15840, 21042, 27610, 36225, 46992, 60900, 78260, 100386, 127820, 162516, 205260, 258819, 324576, 406230, 506022, 629195, 778932, 962555, 1185030

Offset: 1

Omar E. Pol, Jan 30 2011

a(n) is the sum of (the zeroth moments of) all partitions of n minus the partition number of n.

			a(7) = 90 = (7-1)*15 = 105 - 15, because the number of partitions of 7 is 15 and the sum of all parts of all partitions of 7 is 7*15 = 105.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000041, A066186. Column 1 of A182729.

a:= n-> (n-1) *combinat[numbpart](n):
seq (a(n), n =1..50);

pnxt[n_]:=Module[{ps=IntegerPartitions[n]},Total[Flatten[ps]]- Length[ps]]; Array[pnxt,40] (* Harvey P. Dale, Jul 15 2011 *)
Table[(n-1)PartitionsP[n],{n,40}] (* Harvey P. Dale, Jan 17 2015 *)

a(n) = (n-1)*A000041(n) = A066186(n) - A000041(n).

cp's OEIS Frontend

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

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A182727 Sum of largest parts of the shell model of partitions with n regions.

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

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A182724 Sum of all parts of all partitions of n minus the number of partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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