A245099 -id:A245099 - 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

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).

A221530 Triangle read by rows: T(n,k) = A000005(k)*A000041(n-k).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 4, 2, 3, 5, 6, 4, 3, 2, 7, 10, 6, 6, 2, 4, 11, 14, 10, 9, 4, 4, 2, 15, 22, 14, 15, 6, 8, 2, 4, 22, 30, 22, 21, 10, 12, 4, 4, 3, 30, 44, 30, 33, 14, 20, 6, 8, 3, 4, 42, 60, 44, 45, 22, 28, 10, 12, 6, 4, 2, 56, 84, 60, 66, 30, 44, 14, 20, 9, 8, 2, 6

Offset: 1

Omar E. Pol, Jan 19 2013

T(n,k) is the number of partitions of n that contain k as a part multiplied by the number of divisors of k.
It appears that T(n,k) is also the total number of appearances of k in the last k sections of the set of partitions of n multiplied by the number of divisors of k.
T(n,k) is also the number of partitions of k into equal parts multiplied by the number of ones in the j-th section of the set of partitions of n, where j = (n - k + 1).
For another version see A245095. - Omar E. Pol, Jul 15 2014

			For n = 6:
  -------------------------
  k   A000005        T(6,k)
  1      1  *  7   =    7
  2      2  *  5   =   10
  3      2  *  3   =    6
  4      3  *  2   =    6
  5      2  *  1   =    2
  6      4  *  1   =    4
  .         A000041
  -------------------------
So row 6 is [7, 10, 6, 6, 4, 2]. Note that the sum of row 6 is 7+10+6+6+2+4 = 35 equals A006128(6).
.
Triangle begins:
  1;
  1,   2;
  2,   2,  2;
  3,   4,  2,  3;
  5,   6,  4,  3,  2;
  7,  10,  6,  6,  2,  4;
  11, 14, 10,  9,  4,  4,  2;
  15, 22, 14, 15,  6,  8,  2,  4;
  22, 30, 22, 21, 10, 12,  4,  4,  3;
  30, 44, 30, 33, 14, 20,  6,  8,  3,  4;
  42, 60, 44, 45, 22, 28, 10, 12,  6,  4,  2;
  56, 84, 60, 66, 30, 44, 14, 20,  9,  8,  2,  6;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

Similar to A221529.
Columns 1-2: A000041, A139582. Leading diagonals 1-3: A000005, A000005, A062011. Row sums give A006128.
Cf. A027293, A135010, A138137, A182703, A245095, A245099.

A221530row[n_]:=DivisorSigma[0,Range[n]]PartitionsP[n-Range[n]];Array[A221530row,10] (* Paolo Xausa, Sep 04 2023 *)

row(n) = vector(n, i, numdiv(i)*numbpart(n-i)); \\ Michel Marcus, Jul 18 2014

T(n,k) = d(k)*p(n-k) = A000005(k)*A027293(n,k).

A245095 Triangle read by rows: T(n,k) = A006218(k)*A002865(n-k).

Original entry on oeis.org

1, 0, 3, 1, 0, 5, 1, 3, 0, 8, 2, 3, 5, 0, 10, 2, 6, 5, 8, 0, 14, 4, 6, 10, 8, 10, 0, 16, 4, 12, 10, 16, 10, 14, 0, 20, 7, 12, 20, 16, 20, 14, 16, 0, 23, 8, 21, 20, 32, 20, 28, 16, 20, 0, 27, 12, 24, 35, 32, 40, 28, 32, 20, 23, 0, 29, 14, 36, 40, 56, 40, 56, 32, 40, 23, 27, 0, 35

Offset: 1

Omar E. Pol, Jul 14 2014

Row sums give A006128, n >= 1.
Column 1 is A002865.
Leading diagonal is A006218, n >= 1.
For another version see A221530.

			Triangle begins:
  1;
  0,   3;
  1,   0,  5;
  1,   3,  0,  8;
  2,   3,  5,  0, 10;
  2,   6,  5,  8,  0, 14;
  4,   6, 10,  8, 10,  0, 16;
  4,  12, 10, 16, 10, 14,  0, 20;
  7,  12, 20, 16, 20, 14, 16,  0, 23;
  8,  21, 20, 32, 20, 28, 16, 20,  0, 27;
  12, 24, 35, 32, 40, 28, 32, 20, 23,  0, 29;
  14, 36, 40, 56, 40, 56, 32, 40, 23, 27,  0, 35;
  ...
For n = 6:
  -------------------------
  k   A006218        T(6,k)
  -------------------------
  1      1  *  2   =    2
  2      3  *  2   =    6
  3      5  *  1   =    5
  4      8  *  1   =    8
  5     10  *  0   =    0
  6     14  *  1   =   14
  .         A002865
  -------------------------
So row 6 is [2, 6, 5, 8, 0, 14] and the sum of row 6 is 2+6+5+8+0+14 = 35 equaling A006128(6) = 35.

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

Cf. A000005, A000041, A002865, A006128, A006218, A221529, A221530, A245099.

A245095row[n_]:=Accumulate[DivisorSigma[0,Range[n]]]Reverse[Differences[PartitionsP[Range[-1,n-1]]]];Array[A245095row,10] (* Paolo Xausa, Sep 04 2023 *)

a006218(n) = sum(k=1, n, n\k);
a002865(n) = if(n, numbpart(n)-numbpart(n-1), 1);
row(n) = vector(n, i, a006218(i)*a002865(n-i)); \\ Michel Marcus, Jul 18 2014

A245093 Triangle read by rows in which row n lists the first n terms of A000203.

Original entry on oeis.org

1, 1, 3, 1, 3, 4, 1, 3, 4, 7, 1, 3, 4, 7, 6, 1, 3, 4, 7, 6, 12, 1, 3, 4, 7, 6, 12, 8, 1, 3, 4, 7, 6, 12, 8, 15, 1, 3, 4, 7, 6, 12, 8, 15, 13, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28

Offset: 1

Omar E. Pol, Jul 15 2014

Reluctant sequence of A000203.
Row sums give A024916.
Has a symmetric representation - for more information see A237270.

			Triangle begins:
1;
1, 3;
1, 3, 4;
1, 3, 4, 7;
1, 3, 4, 7, 6;
1, 3, 4, 7, 6, 12;
1, 3, 4, 7, 6, 12, 8;
1, 3, 4, 7, 6, 12, 8, 15;
1, 3, 4, 7, 6, 12, 8, 15, 13;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A000203, A024916, A027293, A104765, A140207, A221529, A237270, A237593, A245099, A265652.

import Data.List (inits)
a245093 n k = a245093_tabl !! (n-1) !! (k-1)
a245093_row n = a245093_tabl !! (n-1)
a245093_tabl = tail $ inits $ a000203_list
-- Reinhard Zumkeller, Dec 12 2015

T(n,k) = A000203(k), 1<=k<=n.

A340424 Triangle read by rows: T(n,k) = A024916(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 4, 0, 8, 0, 1, 15, 0, 4, 1, 21, 0, 8, 4, 2, 33, 0, 15, 8, 8, 2, 41, 0, 21, 15, 16, 8, 4, 56, 0, 33, 21, 30, 16, 16, 4, 69, 0, 41, 33, 42, 30, 32, 16, 7, 87, 0, 56, 41, 66, 42, 60, 32, 28, 8, 99, 0, 69, 56, 82, 66, 84, 60, 56, 32, 12, 127, 0, 87, 69, 112, 82, 132, 84, 105, 64, 48, 14

Offset: 1

Omar E. Pol, Jan 07 2021

Conjecture: the sum of row n equals A066186(n), the sum of all parts of all partitions of n.

			Triangle begins:
   1;
   4,  0;
   8,  0,  1;
  15,  0,  4,  1;
  21,  0,  8,  4,  2;
  33,  0, 15,  8,  8,  2;
  41,  0, 21, 15, 16   8,  4;
  56,  0, 33, 21, 30, 16, 16,  4;
  69,  0, 41, 33, 42, 30, 32, 16,  7;
  87,  0, 56, 41, 66, 42, 60, 32, 28,  8;
  99,  0, 69, 56, 82, 66, 84, 60, 56, 32, 12;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *  33   =  33
2      0   *  21   =   0
3      1   *  15   =  15
4      1   *   8   =   8
5      2   *   4   =   8
6      2   *   1   =   2
.           A024916
--------------------------
The sum of row 6 is 33 + 0 + 15 + 8 + 8 + 2 = 66, equaling A066186(6) = 66.

Mirror of A245099.
Columns 1, 3 and 4 are A024916 (partial sums of A000203).
Column 2 gives A000004.
Columns 5 and 6 give A327329.
Columns 7 and 8 give A243980.
Leading diagonal gives A002865.
Cf. A066186.

A340583 Triangle read by rows: T(n,k) = A002865(n-k)*A000203(k), 1 <= k <= n.

Original entry on oeis.org

1, 0, 3, 1, 0, 4, 1, 3, 0, 7, 2, 3, 4, 0, 6, 2, 6, 4, 7, 0, 12, 4, 6, 8, 7, 6, 0, 8, 4, 12, 8, 14, 6, 12, 0, 15, 7, 12, 16, 14, 12, 12, 8, 0, 13, 8, 21, 16, 28, 12, 24, 8, 15, 0, 18, 12, 24, 28, 28, 24, 24, 16, 15, 13, 0, 12, 14, 36, 32, 49, 24, 48, 16, 30, 13, 18, 0, 28

Offset: 1

Omar E. Pol, Jan 15 2021

T(n,k) is the total number of cubic cells added at n-th stage to the right prisms whose bases are the parts of the symmetric representation of sigma(k) in the polycube described in A221529.

Partial sums of column k gives the column k of A221529.

			Triangle begins:
   1;
   0,  3;
   1,  0,  4;
   1,  3,  0,  7;
   2,  3,  4,  0,  6;
   2,  6,  4,  7,  0, 12;
   4,  6,  8,  7,  6,  0,  8;
   4, 12,  8, 14,  6, 12,  0, 15;
   7, 12, 16, 14, 12, 12,  8,  0, 13;
   8, 21, 16, 28, 12, 24,  8, 15,  0, 18;
  12, 24, 28, 28, 24, 24, 16, 15, 13,  0, 12;
  14, 36, 32, 49, 24, 48, 16, 30, 13, 18,  0, 28;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A000203         T(6,k)
--------------------------
1      1   *   2  =    2
2      3   *   2   =   6
3      4   *   1   =   4
4      7   *   1   =   7
5      6   *   0   =   0
6     12   *   1   =  12
.           A002865
--------------------------
The sum of row 6 is 2 + 6 + 4 + 7 + 0 + 12 = 31, equaling A138879(6).

Row sums give A138879.
Column 1 gives A002865.
Diagonals 1, 3 and 4 give A000203.
Diagonal 2 gives A000004.
Diagonals 5 and 6 give A074400.
Diagonals 7 and 8 give A239050.
Diagonal 9 gives A319527.
Diagonal 10 gives A319528.
Cf. A221529 (partial column sums).
Cf. A340426 (mirror).
Cf. A135010, A221530, A245095, A245099, A339278.

A340583[n_, k_] := (PartitionsP[n - k] - PartitionsP[(n - k) - 1])*
   DivisorSigma[1, k];
Table[A340583[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Robert P. P. McKone, Jan 25 2021 *)

cp's OEIS Frontend

A066186 Sum of all parts of 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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A221530 Triangle read by rows: T(n,k) = A000005(k)*A000041(n-k).

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A245095 Triangle read by rows: T(n,k) = A006218(k)*A002865(n-k).

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A245093 Triangle read by rows in which row n lists the first n terms of A000203.

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A340424 Triangle read by rows: T(n,k) = A024916(n-k+1)*A002865(k-1), 1 <= k <= n.

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A340583 Triangle read by rows: T(n,k) = A002865(n-k)*A000203(k), 1 <= k <= n.

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