A220909 -id:A220909 - cp's OEIS frontend

A066186 Sum of all parts of all partitions of n.

0, 1, 4, 9, 20, 35, 66, 105, 176, 270, 420, 616, 924, 1313, 1890, 2640, 3696, 5049, 6930, 9310, 12540, 16632, 22044, 28865, 37800, 48950, 63336, 81270, 104104, 132385, 168120, 212102, 267168, 334719, 418540, 520905, 647172, 800569, 988570, 1216215, 1493520

Offset: 0

Wouter Meeussen, Dec 15 2001

Sum of the zeroth moments of all partitions of n.
Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that any part z is composed of labeled elements of amount 1, i.e., z = 1_1 + 1_2 + ... + 1_z. Then one can take from z a single element in z different ways. E.g., for n=3 to n=2 we have A066186(3) = 9 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [111], [12] --> [111], [12] --> [2], [3] --> 2, [3] --> 2, [3] --> 2. For the unlabeled case, one can take a single element from z in only one way. Then the number of one-element transitions from the integer partitions of n to the partitions of n-1 is given by A000070. E.g., A000070(3) = 4 and for the transition from n=3 to n=2 one has [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. - Thomas Wieder, May 20 2004
Also sum of all parts of all regions of n (Cf. A206437). - Omar E. Pol, Jan 13 2013
From Omar E. Pol, Jan 19 2021: (Start)
Apart from initial zero this is also as follows:
Convolution of A000203 and A000041.
Convolution of A024916 and A002865.
For n >= 1, a(n) is also the number of cells in a symmetric polycube in which the terraces are the symmetric representation of sigma(k), for k = n..1, (cf. A237593) starting from the base and located at the levels A000041(0)..A000041(n-1) respectively. The polycube looks like a symmetric tower (cf. A221529). A dissection is a three-dimensional spiral whose top view is described in A239660. The growth of the volume of the polycube represents each convolution mentioned above. (End)
From Omar E. Pol, Feb 04 2021: (Start)
a(n) is also the sum of all divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned divisors are also all parts of all partitions of n.
Apart from initial zero this is also the convolution of A340793 and A000070. (End)

			a(3)=9 because the partitions of 3 are: 3, 2+1 and 1+1+1; and (3) + (2+1) + (1+1+1) = 9.
a(4)=20 because A000041(4)=5 and 4*5=20.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265, alternate copy. - From _N. J. A. Sloane_, Jan 02 2013
F. G. Garvan, Higher-order spt functions, arXiv:1008.1207 [math.NT], 2010.
T. J. Osler, A. Hassen and T. R. Chandrupatia, Surprising connections between partitions and divisors, The College Mathematics Journal, Vol. 38. No. 4, Sep. 2007, 278-287 (see p. 287).
Omar E. Pol, Illustration of a(10), prism and tower, each polycube contains 420 cubes.
Omar E. Pol, Illustration of initial terms of A066186 and of A139582 (n>=1)

Cf. A000041, A093694, A000070, A132825, A001787 (same for ordered partitions), A277029, A000203, A221529, A237593, A239660.
First differences give A138879. - Omar E. Pol, Aug 16 2013
Row sums of triangles A138785, A181187, A245099, A337209, A339106, A340423, A340424, A221529, A302246, A338156, A340035, A340056, A340057, A346741. - Omar E. Pol, Aug 02 2021

a066186 = sum . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

with(combinat): a:= n-> n*numbpart(n): seq(a(n), n=0..50); # Zerinvary Lajos, Apr 25 2007

PartitionsP[ Range[0, 60] ] * Range[0, 60]

a(n)=numbpart(n)*n \\ Charles R Greathouse IV, Mar 10 2012

from sympy import npartitions
def A066186(n): return n*npartitions(n) # Chai Wah Wu, Oct 22 2023

[n*Partitions(n).cardinality() for n in range(41)] # Peter Luschny, Jul 29 2014

a(n) = n * A000041(n). - Omar E. Pol, Oct 10 2011
G.f.: x * (d/dx) Product_{k>=1} 1/(1-x^k), i.e., derivative of g.f. for A000041. - Jon Perry, Mar 17 2004 (adjusted to match the offset by Geoffrey Critzer, Nov 29 2014)
Equals A132825 * [1, 2, 3, ...]. - Gary W. Adamson, Sep 02 2007
a(n) = A066967(n) + A066966(n). - Omar E. Pol, Mar 10 2012
a(n) = A207381(n) + A207382(n). - Omar E. Pol, Mar 13 2012
a(n) = A006128(n) + A196087(n). - Omar E. Pol, Apr 22 2012
a(n) = A220909(n)/2. - Omar E. Pol, Jan 13 2013
a(n) = Sum_{k=1..n} A000203(k)*A000041(n-k), n >= 1. - Omar E. Pol, Jan 20 2013
a(n) = Sum_{k=1..n} k*A036043(n,n-k+1). - L. Edson Jeffery, Aug 03 2013
a(n) = Sum_{k=1..n} A024916(k)*A002865(n-k), n >= 1. - Omar E. Pol, Jul 13 2014
a(n) ~ exp(Pi*sqrt(2*n/3))/(4*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Oct 24 2016
a(n) = Sum_{k=1..n} A340793(k)*A000070(n-k), n >= 1. - Omar E. Pol, Feb 04 2021

a(0) added by Franklin T. Adams-Watters, Jul 28 2014

A092269 Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n.

Original entry on oeis.org

1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589, 801, 1048, 1407, 1820, 2399, 3087, 3998, 5092, 6545, 8263, 10486, 13165, 16562, 20630, 25773, 31897, 39546, 48692, 59960, 73423, 89937, 109553, 133439, 161840, 196168, 236843, 285816, 343667, 412950, 494702, 592063, 706671

Offset: 1

Vladeta Jovovic, Feb 16 2004

nonn

Row sums of triangle A220504. - Omar E. Pol, Jan 19 2013

			Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4]. 1 appears 4 times in the first, 1 twice in the second, 2 twice in the third, etc.; thus a(4)=4+2+2+1+1=10.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 550 terms from Joerg Arndt)
F. G. Garvan, Table of a(n) for n = 1..10000 (Coefficients of Andrews spt-function)
G. E. Andrews, The number of smallest parts in the partitions of n, Journal für die reine und angewandte Mathematik, Volume 2008 Issue 624, 133-142.
Scott Ahlgren, Nickolas Andersen, Euler-like recurrences for smallest parts functions, arXiv:1402.5366
George E. Andrews, Song Heng Chan and Byungchan Kim, The Odd Moments of Ranks and Cranks, Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91. - From _N. J. A. Sloane_, Sep 04 2012
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function
G. E. Andrews, F. G. Garvan, and J. Liang, Self-conjugate vector partitions and the parity of the spt-function
William Y.C. Chen, The spt-Function of Andrews, arXiv:1707.04369 [math.CO], Jul 14 2017.
A. Folsom and K. Ono, The spt-function of Andrews, PNAS, December 23 2008, 105 (51) 20152-20156.
F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13, arXiv:1011.1955 [math.NT], Nov 9 2010.
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], Nov 9 2010.
F. G. Garvan, Higher Order Spt-functions, Adv. Math. 228 (2011), no. 1, 241-265. - From _N. J. A. Sloane_, Jan 02 2013
F. G. Garvan, The smallest parts partition function, 2012.
F. G. Garvan, Dyson's rank function and Andrews's SPT-function, slides 11, 12.
M. H. Mertens, K. Ono, L. Rolen, Mock modular Eisenstein series with Nebentypus, arXiv:1906.07410 [math.NT], 2019.
K. Ono, Congruences for the Andrews spt-function, PNAS, December 21 2010, 108 (2) 473-476.
Omar E. Pol, Illustration of initial terms
Wikipedia, Spt function

Cf. A092314, A092322, A092309, A092321, A092313, A092310, A092311, A092268, A006128, A195053.

For higher-order spt functions see A221140-A221144.

b:= proc(n, i) option remember; `if`(n=0 or i=1, n,
      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 16 2013

terms = 47; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest (* Jean-François Alcover, Jan 17 2013 *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r==0, q, 0] + Sum[b[n-i*j, i-1], {j, 0, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

N = 66;  x = 'x + O('x^N);
gf = sum(n=1,N, x^n/(1-x^n) * prod(k=n,N, 1/(1-x^k) )  );
v = Vec(gf)
/* Joerg Arndt, Jan 12 2013 */

G.f.: Sum_{n>=1} x^n/(1-x^n) * Product_{k>=n} 1/(1-x^k).
a(n) = A000070(n-1) + A195820(n). - Omar E. Pol, Oct 19 2011
a(n) = n*p(n) - N_2(n)/2 = n*A000041(n) - A220908(n)/2 = A066186(n) - A220907(n) = (A220909(n) - A220908(n))/2 = A211982(n)/2 (from Andrews's paper and Garvan's paper). - Omar E. Pol, Jan 03 2013
a(n) = A000041(n) + A000070(n-2) + A220479(n), n >= 2. - Omar E. Pol, Feb 16 2013
Asymptotics (Bringmann-Mahlburg, 2009): a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*sqrt(8*n)) ~ sqrt(6*n)*A000041(n)/Pi. - Vaclav Kotesovec, Jul 30 2017

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A221529 Triangle read by rows: T(n,k) = A000203(k)*A000041(n-k), 1 <= k <= n.

Original entry on oeis.org

1, 1, 3, 2, 3, 4, 3, 6, 4, 7, 5, 9, 8, 7, 6, 7, 15, 12, 14, 6, 12, 11, 21, 20, 21, 12, 12, 8, 15, 33, 28, 35, 18, 24, 8, 15, 22, 45, 44, 49, 30, 36, 16, 15, 13, 30, 66, 60, 77, 42, 60, 24, 30, 13, 18, 42, 90, 88, 105, 66, 84, 40, 45, 26, 18, 12, 56, 126, 120, 154, 90, 132, 56, 75, 39, 36, 12, 28

Offset: 1

Omar E. Pol, Jan 20 2013

Since A000203(k) has a symmetric representation, both T(n,k) and the partial sums of row n can be represented by symmetric polycubes. For more information see A237593 and A237270. For another version see A245099. - Omar E. Pol, Jul 15 2014
From Omar E. Pol, Jul 10 2021: (Start)
The above comment refers to a symmetric tower whose terraces are the symmetric representation of sigma(i), for i = 1..n, starting from the top. The levels of these terraces are the partition numbers A000041(h-1), for h = 1 to n, starting from the base of the tower, where n is the length of the largest side of the base.
The base of the tower is the symmetric representation of A024916(n).
The height of the tower is equal to A000041(n-1).
The surface area of the tower is equal to A345023(n).
The volume (or the number of cubes) of the tower equals A066186(n).
The volume represents the n-th term of the convolution of A000203 and A000041, that is A066186(n).
Note that the terraces that are the symmetric representation of sigma(n) and the terraces that are the symmetric representation of sigma(n-1) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1].
The tower is an object of the family of the stepped pyramid described in A245092.
T(n,k) can be represented with a set of A237271(k) right prisms of height A000041(n-k) since T(n,k) is the total number of cubes that are exactly below the parts of the symmetric representation of sigma(k) in the tower.
T(n,k) is also the sum of all divisors of all k's that are in the first n rows of triangle A336811, or in other words, in the first A000070(n-1) terms of the sequence A336811. Hence T(n,k) is also the sum of all divisors of all k's in the n-th row of triangle A176206.
The mentioned property is due to the correspondence between divisors and parts explained in A338156: all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.
Therefore the set of all partitions of n >= 1 has an associated tower.
The partial column sums of A340583 give this triangle showing the growth of the structure of the tower.
Note that the convolution of A000203 with any integer sequence S can be represented with a symmetric tower or structure of the same family where its terraces are the symmetric representation of sigma starting from the top and the heights of the terraces starting from the base are the terms of the sequence S. (End)

			Triangle begins:
------------------------------------------------------
    n| k    1   2   3   4   5   6   7   8   9  10
------------------------------------------------------
    1|      1;
    2|      1,  3;
    3|      2,  3,  4;
    4|      3,  6,  4,  7;
    5|      5,  9,  8,  7,  6;
    6|      7, 15, 12, 14,  6, 12;
    7|     11, 21, 20, 21, 12, 12,  8;
    8|     15, 33, 28, 35, 18, 24,  8, 15;
    9|     22, 45, 44, 49, 30, 36, 16, 15, 13;
   10|     30, 66, 60, 77, 42, 60, 24, 30, 13, 18;
...
The sum of row 10 is [30 + 66 + 60 + 77 + 42 + 60 + 24 + 30 + 13 + 18] = A066186(10) = 420.
.
For n = 10 the calculation of the row 10 is as follows:
    k    A000203         T(10,k)
    1       1   *  30   =   30
    2       3   *  22   =   66
    3       4   *  15   =   60
    4       7   *  11   =   77
    5       6   *   7   =   42
    6      12   *   5   =   60
    7       8   *   3   =   24
    8      15   *   2   =   30
    9      13   *   1   =   13
   10      18   *   1   =   18
                 A000041
.
From _Omar E. Pol_, Jul 13 2021: (Start)
For n = 10 we can see below three views of two associated polycubes called here "prism of partitions" and "tower". Both objects contain the same number of cubes (that property is valid for n >= 1).
        _ _ _ _ _ _ _ _ _ _
  42   |_ _ _ _ _          |
       |_ _ _ _ _|_        |
       |_ _ _ _ _ _|_      |
       |_ _ _ _      |     |
       |_ _ _ _|_ _ _|_    |
       |_ _ _ _        |   |
       |_ _ _ _|_      |   |
       |_ _ _ _ _|_    |   |
       |_ _ _      |   |   |
       |_ _ _|_    |   |   |
       |_ _    |   |   |   |
       |_ _|_ _|_ _|_ _|_  |                             _
  30   |_ _ _ _ _        | |                            | | 30
       |_ _ _ _ _|_      | |                            | |
       |_ _ _      |     | |                            | |
       |_ _ _|_ _ _|_    | |                            | |
       |_ _ _ _      |   | |                            | |
       |_ _ _ _|_    |   | |                            | |
       |_ _ _    |   |   | |                            | |
       |_ _ _|_ _|_ _|_  | |                           _|_|
  22   |_ _ _ _        | | |                          |   |  22
       |_ _ _ _|_      | | |                          |   |
       |_ _ _ _ _|_    | | |                          |   |
       |_ _ _      |   | | |                          |   |
       |_ _ _|_    |   | | |                          |   |
       |_ _    |   |   | | |                          |   |
       |_ _|_ _|_ _|_  | | |                         _|_ _|
  15   |_ _ _ _      | | | |                        | |   |  15
       |_ _ _ _|_    | | | |                        | |   |
       |_ _ _    |   | | | |                        | |   |
       |_ _ _|_ _|_  | | | |                       _|_|_ _|
  11   |_ _ _      | | | | |                      | |     |  11
       |_ _ _|_    | | | | |                      | |     |
       |_ _    |   | | | | |                      | |     |
       |_ _|_ _|_  | | | | |                     _| |_ _ _|
   7   |_ _ _    | | | | | |                    |   |     |   7
       |_ _ _|_  | | | | | |                   _|_ _|_ _ _|
   5   |_ _    | | | | | | |                  | | |       |   5
       |_ _|_  | | | | | | |                 _| | |_ _ _ _|
   3   |_ _  | | | | | | | |               _|_ _|_|_ _ _ _|   3
   2   |_  | | | | | | | | |           _ _|_ _|_|_ _ _ _ _|   2
   1   |_|_|_|_|_|_|_|_|_|_|          |_ _|_|_|_ _ _ _ _ _|   1
.
             Figure 1.                       Figure 2.
         Front view of the                 Lateral view
        prism of partitions.               of the tower.
.
.                                      _ _ _ _ _ _ _ _ _ _
                                      |   | | | | | | | |_|   1
                                      |   | | | | | |_|_ _|   2
                                      |   | | | |_|_  |_ _|   3
                                      |   | |_|_    |_ _ _|   4
                                      |   |_ _  |_  |_ _ _|   5
                                      |_ _    |_  |_ _ _ _|   6
                                          |_    | |_ _ _ _|   7
                                            |_  |_ _ _ _ _|   8
                                              |           |   9
                                              |_ _ _ _ _ _|  10
.
                                             Figure 3.
                                             Top view
                                           of the tower.
.
Figure 1 is a two-dimensional diagram of the partitions of 10 in colexicographic order (cf. A026792, A211992). The area of the diagram is 10*42 = A066186(10) = 420. Note that the diagram can be interpreted also as the front view of a right prism whose volume is 1*10*42 = 420 equaling the volume and the number of cubes of the tower that appears in the figures 2 and 3.
Note that the shape and the area of the lateral view of the tower are the same as the shape and the area where the 1's are located in the diagram of partitions. In this case the mentioned area equals A000070(10-1) = 97.
The connection between these two associated objects is a representation of the correspondence divisor/part described in A338156. See also A336812.
The sum of the volumes of both objects equals A220909.
For the connection with the table of A338156 see also A340035. (End)

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened)
T. J. Osler, A. Hassen and T. R. Chandrupatia, Surprising connections between partitions and divisors, The College Mathematics Journal, Vol. 38. No. 4, Sep. 2007, 278-287 (see p. 287).
Omar E. Pol, Illustration of the prism, the tower and the 10th row of the triangle

Row sums give A066186.
Column 1 is A000041.
Leading diagonals 1-5: A000203, A000203, A074400, A272027, A274535.
Companion of A221530.
Cf. A000070, A000203, A026792, A027293, A135010, A138137, A176206, A182703, A220909, A211992, A221649, A236104, A237270, A237271, A237593, A245092, A245093, A245095, A245099, A262626, A336811, A336812, A338156, A339278, A340035, A340583, A340584, A345023, A346741.

nrows=12; Table[Table[DivisorSigma[1,k]PartitionsP[n-k],{k,n}],{n,nrows}] // Flatten (* Paolo Xausa, Jun 17 2022 *)

T(n,k)=sigma(k)*numbpart(n-k) \\ Charles R Greathouse IV, Feb 19 2013

T(n,k) = sigma(k)*p(n-k) = A000203(k)*A027293(n,k).

T(n,k) = A245093(n,k)*A027293(n,k).

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

A211978 Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n.

Original entry on oeis.org

0, 2, 6, 12, 24, 40, 70, 108, 172, 256, 384, 550, 798, 1112, 1560, 2136, 2926, 3930, 5288, 6996, 9260, 12104, 15798, 20412, 26348, 33702, 43044, 54588, 69090, 86906, 109126, 136270, 169854, 210732, 260924, 321752, 396028, 485624, 594402, 725174, 883092, 1072208

Offset: 0

Omar E. Pol, Jan 03 2013

nonn

Also twice A006128, because the total number of parts in all partitions of n equals the sum of largest parts of all partitions of n. For a proof without words see the illustration of initial terms. Note that the sum of the lengths of all horizontal segments equals the sum of largest parts of all partitions of n. On the other hand, the sum of the lengths of all vertical segments equals the total number of parts of all partition of n. Therefore the sum of lengths of all horizontal segments equals the sum of lengths of all vertical segments.
a(n) is also the sum of the semiperimeters of the Ferrers boards of the partitions of n. Example: a(2)=6; indeed, the Ferrers boards of the partitions [2] and [1,1] of 2 are 2x1 rectangles; the sum of their semiperimeters is 3 + 3 = 6. - Emeric Deutsch, Oct 07 2016
a(n) is also the sum of the semiperimeters of the regions of the set of partitions of n. See the first illustration in the Example section. For more information see A278355. - Omar E. Pol, Nov 23 2016

			Illustration of initial terms as a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                         _ _ _      |
.                                         _ _ _|_    |
.                                         _ _    |   |
.                             _ _ _ _ _   _ _|_ _|_  |
.                             _ _ _    |  _ _ _    | |
.                   _ _ _ _   _ _ _|_  |  _ _ _|_  | |
.                   _ _    |  _ _    | |  _ _    | | |
.           _ _ _   _ _|_  |  _ _|_  | |  _ _|_  | | |
.     _ _   _ _  |  _ _  | |  _ _  | | |  _ _  | | | |
. _   _  |  _  | |  _  | | |  _  | | | |  _  | | | | |
.  |   | |   | | |   | | | |   | | | | |   | | | | | |
.
. 2    6     12        24         40          70
.
Also using the elements from the diagram we can draw an infinite Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n) as shown below:
.
11...........................................................
.                                                           /\
.                                                          /  \
.                                                         /    \
7..................................                      /      \
.                                 /\                    /        \
5....................            /  \                /\/          \
.                   /\          /    \          /\  /              \
3..........        /  \        /      \        /  \/                \
2.....    /\      /    \    /\/        \      /                      \
1..  /\  /  \  /\/      \  /            \  /\/                        \
0 /\/  \/    \/          \/              \/                            \
. 0,2,  6,   12,         24,             40,                          70...
.

Michael De Vlieger, Table of n, a(n) for n = 0..3000
Toufik Mansour, Armend Sh. Shabani, Enumerations on bargraphs, Discrete Math. Lett. (2019) Vol. 2, 65-94.
Omar E. Pol, Visualization of regions in a diagram for A006128
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139582 is twice A000041. A220909 is twice A066186.

Cf. A006128, A135010, A141285, A186114, A193870, A187219, A194446, A194447, A206437, A211026, A220517, A225600, A278355.

Q := sum(x^j/(1-x^j), j = 1 .. i): R := product(1-x^j, j = 1 .. i): g := sum(x^i*(1+i+Q)/R, i = 1 .. 100): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 41); # Emeric Deutsch, Oct 07 2016

Array[2 Sum[DivisorSigma[0, m] PartitionsP[# - m], {m, #}] &, 42, 0] (* Michael De Vlieger, Mar 20 2020 *)

a(n) = 2*A006128(n).
a(n) = A225600(2*A000041(n)) = A225600(A139582(n)), n >= 1.
a(n) = (Sum_{m=1..p(n)} A194446(m)) + (Sum_{m=1..p(n)} A141285(m)) = 2*Sum_{m=1..p(n)} A194446(m) = 2*Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1.
The trivariate g.f. G(t,s,x) of the partitions of a nonnegative integer relative to weight (marked by x), number of parts (marked by t), and largest part (marked by s) is G(t,s,x) = Sum_{i>=1} t*s^i*x^i/product_{j=1..i} (1-tx^j). Setting s = t, we obtain the bivariate g.f. of the partitions relative to weight (marked by x) and semiperimeter of the Ferrers board (marked by t). The g.f. of a(n) is g(x) = Sum_{i>=1} ((x^i*(1 + i + Q(x))/R(x)), where Q(x) = sum_{j=1..i} (x^j/(1 - x^j)) and R(x) = product_{j=1..i}(1-x^j). g(x) has been obtained by setting t = 1 in dG(t,t,x))/dt. - Emeric Deutsch, Oct 07 2016

A220908 The second rank moment function N_2(n).

Original entry on oeis.org

0, 2, 8, 20, 42, 80, 140, 238, 380, 602, 910, 1372, 1996, 2900, 4102, 5790, 8002, 11046, 14980, 20282, 27090, 36092, 47546, 62510, 81374, 105700, 136210, 175084, 223510, 284694, 360410, 455244, 572054, 717160, 894964, 1114470, 1382032, 1710262, 2108750, 2594704, 3182120

Offset: 1

N. J. A. Sloane, Jan 02 2013

nonn

N_2(n) is also called the second Atkin-Garvan moment (see Andrews' paper). - Omar E. Pol, Oct 23 2013

Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews, The number of smallest parts in the partitions of n, p. 3
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265. See Eq. (1.1).

Cf. A000041, A092269, A220907.

b:= proc(n, i) option remember; `if`(n=0 or i<1, 0,
      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> 2*(n*combinat[numbpart](n)- b(n, n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 09 2013

terms = 41; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; spt = CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest; a[n_] := 2*(n*PartitionsP[n] - spt[[n]]); Table[a[n], {n, 1, terms}] (* Jean-François Alcover, Jan 17 2013, after g.f. of spt(n) *)

a(n) = 2*A220907(n) = 2*(n*A000041(n)-A092269(n)).
a(n) = 2*(A066186(n) - A092269(n)). - Omar E. Pol, Jan 09 2013
a(n) = A220909(n) - A211982(n). - Omar E. Pol, Jan 16 2013
a(n) ~ exp(Pi*sqrt(2*n/3))/(2*sqrt(3)) * (1 - (3*sqrt(6)/(2*Pi) + Pi/(24*sqrt(6)))/sqrt(n) + (5/48 + Pi^2/6912)/n). - Vaclav Kotesovec, Jul 31 2017

A183009 a(n) = 24np(n) = 24nA000041(n).

Original entry on oeis.org

24, 96, 216, 480, 840, 1584, 2520, 4224, 6480, 10080, 14784, 22176, 31512, 45360, 63360, 88704, 121176, 166320, 223440, 300960, 399168, 529056, 692760, 907200, 1174800, 1520064, 1950480, 2498496, 3177240, 4034880, 5090448, 6412032

Offset: 1

Omar E. Pol, Jan 22 2011

a(n) is also the sum of the partition number of n and the "trace" Tr(n) of A183011. a(n) = p(n) + Tr(n).

a(n) is also the number of "sectors" or "half-periods" in all partitions of n in some versions of the shell model of partitions of A135010.

			The number of partitions of 6 is p(6) = A000041(6) = 11, so a(6) = 24*6*11 = 1584.
Also the trace Tr(6) = A183011(6) = 1573, so a(6) = p(6) + Tr(6) = 11 + 1573 = 1584.

Partial sums of A183006. Column 24 of A182728.

Cf. A000041, A000796, A008606, A018253, A066186, A135010, A182742, A182743, A183008, A183010, A183011.

Table[24n*PartitionsP[n],{n,40}] (* Harvey P. Dale, Mar 07 2019 *)

a(n) = A008606(n)*A000041(n) = 24*A066186(n) = n*A183008(n).
a(n) = p(n) + Tr(n) = A000041(n) + A183011(n).
a(n) = 12*M_2(n) = 24*spt(n) + 12*N_2(n) = 12*A220909(n) = 24*A092269(n) + 12*A220908(n). - Omar E. Pol, Feb 17 2013

A211982 Second crank moment minus second rank moment: M_2(n) - N_2(n) = 2*spt(n).

Original entry on oeis.org

2, 6, 10, 20, 28, 52, 70, 114, 160, 238, 322, 476, 630, 880, 1178, 1602, 2096, 2814, 3640, 4798, 6174, 7996, 10184, 13090, 16526, 20972, 26330, 33124, 41260, 51546, 63794, 79092, 97384, 119920, 146846, 179874, 219106, 266878, 323680, 392336, 473686

Offset: 1

Omar E. Pol, Jan 03 2013

nonn

Also total number of smallest parts in all partitions of n, multiplied by 2.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews, The number of smallest parts in the partitions of n
F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
F. G. Garvan, Higher order spt-functions

Cf. A092269, A066186, A220907, A220908, A220909.

b:= proc(n, i) option remember; `if`(n=0 or i=1, n,
      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> 2* b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 17 2013

terms = 41; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; 2*CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest (* Jean-François Alcover, Jan 17 2013, from 2nd formula *)

a(n) = A220909(n) - A220908(n) = 2*A092269(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*sqrt(2*n)) * (1 - Pi/(24*sqrt(6*n)) + (144+Pi^2)/(6912*n)). - Vaclav Kotesovec, Jul 31 2017

A350333 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the partitions of n in the following order: row n lists the n-th row of A026792 followed by the n-th row of A338156.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 3, 2, 4, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2, 4, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 6, 3, 3, 4, 2, 2, 2, 2, 5, 1

Offset: 1

Omar E. Pol, Dec 25 2021

			Triangle begins:
[1], [1];
[2, 1, 1], [1, 2, 1];
[3, 2, 1, 1, 1, 1], [1, 3, 1, 2, 1, 1];
[4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1], [1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1];
...
Illustration of the first six rows of triangle in an infinite table:
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| n |         |  1  |   2   |    3    |     4     |      5      |       6       |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   |         |     |       |         |           |             |  6            |
| P |         |     |       |         |           |             |  3 3          |
| A |         |     |       |         |           |             |  4 2          |
| R |         |     |       |         |           |             |  2 2 2        |
| T |         |     |       |         |           |  5          |  5 1          |
| I |         |     |       |         |           |  3 2        |  3 2 1        |
| T |         |     |       |         |  4        |  4 1        |  4 1 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |  2 2 1 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |  3 1 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |  2 1 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |  1 1 1 1 1 1  |
----|---------|-----|-------|---------|-----------|-------------|---------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |  1 2 3     6  |
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |  1       5    |
|   | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
|   | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
|   | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| D | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| I | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| V | A027750 |     |       |         |           |  1          |  1 2          |
| I | A027750 |     |       |         |           |  1          |  1 2          |
| S | A027750 |     |       |         |           |  1          |  1 2          |
| O | A027750 |     |       |         |           |  1          |  1 2          |
| R | A027750 |     |       |         |           |  1          |  1 2          |
| S | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
For n = 6 in the upper zone of the above table we can see the partitions of 6 in reverse-colexicographic order in accordance with the 6th row of A026792.
In the lower zone of the table we can see the terms from the 6th row of A338156, these are the divisors of the numbers from the 6th row of A176206.
Note that in the lower zone of the table every row gives A027750.
The total number of rows in the table is equal to A000070(6+1) = 30.
The remarkable fact is that the elements in the lower zone of the arrangement are the same as the elements in the upper zone but in other order.
For an explanation of the connection of the elements of the upper zone with the elements of the lower zone, that is the correspondence divisor/part, see A338156.
For n = 10 we can see a representation of the upper zone (the partitions) and of the lower zone (the divisors) with the two polycubes described in A221529 respectively: a prism of partitions and a tower whose terraces are the symmetric representation of sigma(m), for m = 1..10. Each polycube has A066186(10) = 420 cubic cells, hence the total number of cubic cells is equal to A220909(10) = 840, equaling the sum of the 10th row of this triangle.

Row sums give A220909.
Row lengths give A211978.
Cf. A350357 (analog for the last section of the set of partitions of n).
Cf. A000041, A000070, A006128, A026792, A027750, A066186, A135010, A176206, A221529, A237593, A336811, A336812, A338156, A340035.

A211609 12 times the total number of smallest parts in all partitions of n, with a(0) = 0.

Original entry on oeis.org

0, 12, 36, 60, 120, 168, 312, 420, 684, 960, 1428, 1932, 2856, 3780, 5280, 7068, 9612, 12576, 16884, 21840, 28788, 37044, 47976, 61104, 78540, 99156, 125832, 157980, 198744, 247560, 309276, 382764, 474552, 584304, 719520, 881076, 1079244, 1314636, 1601268, 1942080, 2354016, 2842116

Offset: 0

Omar E. Pol, Feb 16 2013

nonn

The product 12spt(n) appears in the formula b(n) = 12spt(n)+(24n-1)p(n) which is mentioned in several papers (see Ono's paper, see also Garvan's papers and Garvan's slides in link section). Note that b(n) is A220481(n).

Observation: first 13 terms coincide with the differences between all terms mentioned in a table of special mock Jacobi forms and the first 13 terms of A183011. For the table see Dabholkar-Murthy-Zagier paper, appendix A.1, table of Q_M (weight 2 case), M = 6, C_M = 12. See also the table in page 46. Question: do all terms coincide?

Atish Dabholkar, Sameer Murthy, Don Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074 [hep-th], 2012-2014.
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
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), (2.12).
F. G. Garvan, The smallest parts partition function, slides, 2012
Ken Ono, Congruences for the Andrews spt-function, PNAS January 11, 2011 108 (2) 473-476.

Cf. A000041, A092269, A183010, A183011, A211982, A220481, A220908, A220909.

a(n) = A220481(n) - A183011(n).

a(n) = 12spt(n) = 12*A092269(n) = 6(M_2(n) - N_2(n)) = 6*A211982(n) = 6*(A220909(n) - A220908(n)), n >= 1.

cp's OEIS Frontend

A066186 Sum of all parts of all partitions of n.

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A092269 Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n.

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A221529 Triangle read by rows: T(n,k) = A000203(k)*A000041(n-k), 1 <= k <= n.

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

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A211978 Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n.

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A220908 The second rank moment function N_2(n).

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A183009 a(n) = 24*n*p(n) = 24*n*A000041(n).

A183009 a(n) = 24np(n) = 24nA000041(n).