A182703 -id:A182703 - cp's OEIS frontend

A340056 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 4, 1, 3, 2, 4, 3, 1, 5, 1, 2, 4, 2, 6, 3, 6, 5, 1, 2, 3, 6, 1, 5, 2, 4, 8, 3, 9, 5, 10, 7, 1, 7, 1, 2, 3, 6, 2, 10, 3, 6, 12, 5, 15, 7, 14, 11, 1, 2, 4, 8, 1, 7, 2, 4, 6, 12, 3, 15, 5, 10, 20, 7, 21, 11, 22, 15, 1, 3, 9, 1, 2, 4, 8, 2, 14, 3, 6, 9, 18, 5

Offset: 1

Omar E. Pol, Dec 27 2020

This triangle is a condensed version of the more irregular triangle A338156 which is the main sequence with further information about the correspondence divisor/part.

			Triangle begins:
  [1];
  [1, 2],    [1];
  [1, 3],    [1, 2],    [2];
  [1, 2, 4], [1, 3],    [2, 4], [3];
  [1, 5],    [1, 2, 4], [2, 6], [3, 6], [5];
  [...
The row sums of triangle give A066186.
Written as an irregular tetrahedron the first five slices are:
  1;
  -----
  1, 2,
  1;
  -----
  1, 3,
  1, 2,
  2;
  --------
  1, 2, 4,
  1, 3,
  2, 4,
  3;
  --------
  1, 5,
  1, 2, 4,
  2, 6,
  3, 6,
  5;
  --------
The row sums of tetrahedron give A339106.
The slices of the tetrahedron appear in the following table formed by four zones shows the correspondence between divisor and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |     |       |         |           |             |
| A |         |     |       |         |           |             |
| R |         |     |       |         |           |             |
| T |         |     |       |         |           |  5          |
| I |         |     |       |         |           |  3 2        |
| T |         |     |       |         |  4        |  4 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A027750 |     |       |  1      |  1 2      |  1   3      |
| I | A027750 |     |       |  1      |  1 2      |  1   3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A027750 |     |       |         |  1        |  1 2        |
| S | A027750 |     |       |         |  1        |  1 2        |
| O | A027750 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
| C | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
| O |    -    |     |       |  2      |  2 4      |  2   6      |
| N |    -    |     |       |         |  3        |  3 6        |
| D |    -    |     |       |         |           |  5          |
|---|---------|-----|-------|---------|-----------|-------------|
.
The lower zone is a condensed version of the "divisors" zone.

Paolo Xausa, Table of n, a(n) for n = 1..11528 (rows 1..75 of the triangle, flattened)

Nonzero terms of A340011.
Row sums give A066186.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340061.

A340056row[n_]:=Flatten[Table[Divisors[n-m]PartitionsP[m],{m,0,n-1}]];Array[A340056row,10] (* Paolo Xausa, Sep 01 2023 *)

A206555 Number of 5's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 4, 5, 8, 10, 15, 18, 26, 32, 44, 56, 73, 92, 120, 149, 193, 238, 302, 373, 469, 576, 716, 876, 1081, 1316, 1615, 1954, 2383, 2875, 3483, 4188, 5048, 6043, 7253, 8653, 10341, 12293, 14634, 17340, 20567, 24300, 28717, 33830

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024789. Also number of occurrences of 5 in all partitions of n that do not contain 1 as a part. It appears that the sum of five successive terms gives the partition numbers A000041 (see A182703 and A194812).

Column 5 of A182703 and of A194812.

Cf. A135010, A138121, A182712-A182714.

A206555 = lambda n: sum(list(p).count(5) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..5} a(n+j), n >= 0.

More terms from Alois P. Heinz, Feb 20 2012

A206560 Number of 10's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 14, 22, 25, 36, 43, 59, 70, 95, 113, 150, 179, 232, 278, 356, 426, 537, 644, 803, 960, 1189, 1417, 1739, 2072, 2523, 2999, 3631, 4304, 5181, 6130, 7342, 8662, 10330, 12159, 14437, 16958

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024794. Also number of occurrences of 10 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of ten successive terms give the partition numbers A000041.

Column 10 of A182703 and of A194812.

Cf. A000041, A135010, A138121, A182712-A182714, A206555-A206559.

A206560 = lambda n: sum(list(p).count(10) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..10} a(n+j), n >= 0.

A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

Original entry on oeis.org

0, 3, 10, 21, 42, 69, 123, 189, 304, 458, 693, 998, 1474, 2067, 2927, 4056, 5613, 7595, 10335, 13782, 18411, 24276, 31944, 41583, 54152, 69762, 89758, 114668, 146181, 185083, 234051, 294126, 368992, 460669, 573906, 711865, 881506, 1087023, 1338043

Offset: 0

Omar E. Pol, Apr 21 2012

nonn

Each part is represented by a cuboid 1 X 1 X L where L is the size of the part.

			For n = 7 the shadows of the three views of the shell model of partitions version "tree" with seven shells looks like this:
.                                        |  Partitions
.    A194805(7) = 25    A066186(7) = 105 |  of 7
.                                        |
.                   1    * * * * * * 1   |  7
.                 2      * * * 1 * * 2   |  4+3
.               2        * * * * 1 * 2   |  5+2
.             3          * * 1 * 2 * 3   |  3+2+2
.   1       2            * * * * * 1 2   |  6+1
.     2     3            * * 1 * * 2 3   |  3+3+1
.       2   3            * * * 1 * 2 3   |  4+2+1
.         3 4            * 1 * 2 * 3 4   |  2+2+2+1
.           3   1        * * * * 1 2 3   |  5+1+1
.           4 2          * * 1 * 2 3 4   |  3+2+1+1
.       1   4            * * * 1 2 3 4   |  4+1+1+1
.         2 5            * 1 * 2 3 4 5   |  2+2+1+1+1
.           5 1          * * 1 2 3 4 5   |  3+1+1+1+1
.         1 6            * 1 2 3 4 5 6   |  2+1+1+1+1+1
.           7            1 2 3 4 5 6 7   |  1+1+1+1+1+1+1
.   ----------------------------------   |
.                                        |
.   * * * * 1 * * * *                    |
.   * * * 1 2 * * * *                    |
.   * 1 * * 2 1 * * *                    |
.   * * 1 2 2 * * 1 *                    |
.   * * * * 2 2 1 * *                    |
.   1 2 2 3 2 * * * *                    |
.           2 3 2 2 1                    |
.                                        |
.    A194804(7) = 59                     |
.
Note that, as a variant, in this case each part is labeled with its position in the partition.
The areas of the shadows of the three views are A066186(7) = 105, A194804(7) = 59 and A194805(7) = 25, therefore the total area of the three shadows is 105+59+25 = 189, so a(7) = 189.

Other versions: A207380, A210970, A210979, A210990, A210991.

Cf. A000041, A066186, A135010, A138121, A141285, A182703, A194804, A194805, A206437.

a(n) = A066186(n) + A194804(n) + A194805(n), n >= 1.

A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

Original entry on oeis.org

0, 3, 9, 18, 21, 35, 39, 58, 61, 67, 71, 99, 103, 110, 115, 152, 155, 161, 165, 175, 181, 186, 238, 242, 249, 254, 265, 269, 277, 283, 352, 355, 361, 365, 375, 381, 386, 401, 406, 415, 422, 428, 522, 526, 533, 538, 549, 553, 561, 567, 584, 590, 595, 606

Offset: 0

Omar E. Pol, Apr 30 2012

nonn

It appears that if n is a partition number A000041 then the rotated structure with n regions shows each row as a partition of k such that A000041(k) = n (see example).

For the definition of "regions of n" see A206437.

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.     A182181(11) = 35           A210692(11) = 29
.
.   1                                       1
.   1                                       1
.   1                                       1
.   1                                       1
.   1       1                             1 1
.   1       1                             1 1
.   1       1   1                       1 1 1
.   2       1   1                       1 1 2
.   2       1   1   1                 1 1 1 2
.   3   2   2   2   1 1             1 1 2 2 3
.   6 3 4 2 5 3 4 2 3 2 1         1 2 3 4 5 6
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.
.                                A182727(11) = 35
.
The areas of the shadows of the three views are A182181(11) = 35, A182727(11) = 35 and A210692(11) = 29, therefore the total area of the three shadows is 35+35+29 = 99, so a(11) = 99.
Since n = 11 is a partition number A000041 we can see that the rotated structure with 11 regions shows each row as a partition of 6 because A000041(6) = 11. See below:
.
.                      6
.                    3   3
.                  4       2
.                2   2       2
.              5               1
.            3   2               1
.          4       1               1
.        2   2       1               1
.      3       1       1               1
.    2   1       1       1               1
.  1   1   1       1       1               1
.

Other versions: A207380, A210970, A210979, A210980, A210990.

Cf. A000041, A026905, A135010, A138121, A141285, A182703, A194446, A182181, A182727, A186114, A206437, A210692.

a(n) = A182181(n) + A182727(n) + A210692(n).

a(A000041(n)) = 2*A006128(n) + A026905(n).

A134869 Row sums of triangle A134868.

Original entry on oeis.org

1, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379, 1432

Offset: 1

Gary W. Adamson, Nov 14 2007

Where records occur in A182703. - Omar E. Pol, Feb 14 2012
Consider quadratic polynomials x^2+cx+d. Then a(n) is the number of these polynomials with 0 <= c < n, 0 <= d < n where no polynomial can be horizontally translated into another. For example, a(3) = 7, the coefficients are as follows: (c, d) = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0)}. Two polynomials are excluded, namely x^2+2x+1 = (x+1)^2+0(x+1)+0, and x^2+2x+2 = (x+1)^2+0(x+1)+1. - Griffin N. Macris, Jul 19 2016
a(n) gives the number of regions into which the square [0,1]x[0,1] is divided by the Bernstein polynomials of degree n. - Franck Maminirina Ramaharo, Feb 28 2018

			a(4) = 11 = sum of row 4 terms of triangle A134868: (2, + 2 + 3 + 4).
a(4) = 11 = 1 + 10, where 10 = T(4).
a(4) = 11 = (1, 3, 3, 1) dot (1, 3, 0, 1) = (1 + 9 + 0 + 1).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A134868, A182703.

Essentially the same as A000124.

a:=n->sum((stirling2(j+1,n)), j=1..n):seq(a(n), n=1..50); # Zerinvary Lajos, Apr 12 2008

Table[(n^2 + n)/2 + Boole[n != 1], {n, 53}] (* or *)
Table[PolygonalNumber@ n + Boole[n != 1], {n, 53}] (* Version 10.4, or *)
Table[Sum[StirlingS2[k + 1, n], {k, n}], {n, 53}] (* or *)
Rest@ CoefficientList[Series[x (1 + x - 2 x^2 + x^3)/(1 - x)^3, {x, 0, 53}], x] (* Michael De Vlieger, Jul 19 2016 *)

a(n)=if(n>1, n*(n+1)/2+1, 1) \\ Charles R Greathouse IV, Aug 05 2016

a(n) = 1, then for n>1, a(n) = T(n) + 1, where A000217 = (1, 3, 6, 10, 15, ...).
Binomial transform of [1, 3, 0, 1, -1, 1, -1, 1, ...].
From R. J. Mathar, Oct 27 2008: (Start)
G.f.: x(1+x-2x^2+x^3)/(1-x)^3.
a(n) = 1 + A000217(n) = A000124(n), n > 1. (End)
Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) - 1/2. - Amiram Eldar, Jun 02 2025

A206562 Triangle read by rows: T(n,k) = sum of all parts >= k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 3, 2, 5, 3, 3, 11, 8, 4, 4, 15, 10, 8, 5, 5, 31, 24, 16, 10, 6, 6, 39, 28, 22, 16, 12, 7, 7, 71, 56, 40, 31, 19, 14, 8, 8, 94, 72, 58, 40, 32, 22, 16, 9, 9, 150, 120, 90, 72, 52, 37, 25, 18, 10, 10, 196, 154, 124, 94, 74, 54, 42, 28, 20, 11, 11

Offset: 1

Omar E. Pol, Feb 15 2012

			Triangle begins:
1;
3,   2;
5,   3,  3;
11,  8,  4,  4;
15, 10,  8,  5,  5;
31, 24, 16, 10,  6,  6;
39, 28, 22, 16, 12,  7,  7;
71, 56, 40, 31, 19, 14,  8,  8;
94, 72, 58, 40, 32, 22, 16,  9,  9;

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-2 give A138879, A138880. Diagonal is A000027.

Cf. A135010, A138121, A182703, A206561, A207031, A207032.

A207379 Triangle read by rows: T(n,k) = number of parts that are in the k-th column of the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 4, 4, 3, 2, 1, 1, 4, 4, 4, 3, 2, 1, 1, 7, 7, 6, 5, 3, 2, 1, 1, 8, 8, 8, 6, 5, 3, 2, 1, 1, 12, 12, 11, 10, 7, 5, 3, 2, 1, 1, 14, 14, 14, 12, 10, 7, 5, 3, 2, 1, 1, 21, 21, 20, 18, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Mar 10 2012

Note that for n >= 2 the tail of the last section of n starts at the second column and the second column contains only one part of size 1, thus both the first and the second columns contain the same number of parts. For more information see A135010 and A182703.

			Illustration of initial terms. First six rows of triangle as numbers of parts in the columns from the last sections of the first six natural numbers:
.                                       6
.                                       3 3
.                                       4 2
.                                       2 2 2
.                           5             1
.                           3 2             1
.                 4           1             1
.                 2 2           1             1
.         3         1           1             1
.   2       1         1           1             1
1     1       1         1           1             1
---------------------------------------------------
1,  1,1,  1,1,1,  2,2,1,1,  2,2,2,1,1,  4,4,3,2,1,1
...
Triangle begins:
1;
1,   1;
1,   1,  1;
2,   2,  1,  1;
2,   2,  2,  1,  1;
4,   4,  3,  2,  1,  1;
4,   4,  4,  3,  2,  1,  1;
7,   7,  6,  5,  3,  2,  1,  1;
8,   8,  8,  6,  5,  3,  2,  1,  1;
12, 12, 11, 10,  7,  5,  3,  2,  1,  1;
14, 14, 14, 12, 10,  7,  5,  3,  2,  1,  1;
21, 21, 20, 18, 14, 11,  7,  5,  3,  2,  1,  1;

Column 1 is A187219. Row sums give A138137. Reversed rows converge to A000041.

Cf. A002865, A058399, A135010, A138135, A181187, A207031, A207380.

A209655 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 4, 1, 2, 2, 1, 1, 2, 1, 1, 7, 6, 1, 4, 2, 1, 2, 3, 1, 1, 1, 2, 2, 1, 1

Offset: 1

Omar E. Pol, Mar 25 2012

Each slice of the tetrahedron is a triangle, thus the number of elements in the n-th slice is A000217(n). The slices are perpendicular to the slices of A026792. Each element of the n-th slice equals the volume of a column of the shell model of partitions with n shells. The sum of each row of the n-th slice is A000041(n). The sum of all elements of the n-th slice is A066186(n).
It appears that the triangle formed by the last row of each slice gives A008284 and A058398.
It appears that the triangle formed by the first column of each slice gives A058399.
Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k,j) is also the number of k-th parts of all partitions of n in the j-th column of rectangle.

			--------------------------------------------------------
Illustration of first five
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    2,                                            2
.    1, 1,                                         2
.          3,                                      3
.          2, 1,                                   3
.          1, 1, 1,                                3
.                   5,                             5
.                   4, 1,                          5
.                   2, 2, 1,                       5
.                   1, 2, 1, 1,                    5
.                               7,                 7
.                               6, 1,              7
.                               4, 2, 1,           7
.                               2, 3, 1, 1,        7
.                               1, 2, 2, 1, 1,     7
--------------------------------------------------------
. 1, 3, 1, 6, 2, 1,12, 5, 2, 1,20, 8, 4, 2, 1,
.
Written as a triangle begins:
1;
2, 1, 1;
3, 2, 1, 1, 1, 1;
5, 4, 1, 2, 2, 1, 1, 2, 1, 1;
7, 6, 1, 4, 2, 1, 2, 3, 1, 1, 1, 2, 2, 1, 1;
In which row sums give A066186.

Column sums give A181187. Main diagonal gives A210765. Another version is A209918.

Cf. A000041, A000217, A002260, A004736, A008284, A026792, A058398, A058399, A066186, A135010, A182703, A182715, A207380.

A210941 Triangle read by rows in which row n lists the parts > 1 of the n-th zone of the shell model of partitions, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 2, 4, 3, 2, 5, 2, 2, 2, 4, 2, 3, 3, 6, 3, 2, 2, 5, 2, 4, 3, 7, 2, 2, 2, 2, 4, 2, 2, 3, 3, 2, 6, 2, 5, 3, 4, 4, 8, 3, 2, 2, 2, 5, 2, 2, 4, 3, 2, 7, 2, 3, 3, 3, 6, 3, 5, 4, 9, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 2, 5, 3, 2, 4, 4, 2, 8, 2

Offset: 1

Omar E. Pol, Apr 18 2012

The n-th zone of the shell model of partitions is formed by the parts of row n followed by infinitely many parts of size 1 (see example and also A210943).

Row n lists the largest part and the parts > 1 of the n-th zone of the model.

			Triangle                First 15 zones of the
begins                  shell model of partitions
--------------------------------------------------
1;                      1 1 1 1 1 1 1 1 1 1 1...
2;                      . 2 1 1 1 1 1 1 1 1 1...
3;                      . . 3 1 1 1 1 1 1 1 1...
2, 2;                   . 2 . 2 1 1 1 1 1 1 1...
4;                      . . . 4 1 1 1 1 1 1 1...
3, 2;                   . . 3 . 2 1 1 1 1 1 1...
5;                      . . . . 5 1 1 1 1 1 1...
2, 2, 2;                . 2 . 2 . 2 1 1 1 1 1...
4, 2;                   . . . 4 . 2 1 1 1 1 1...
3, 3;                   . . 3 . . 3 1 1 1 1 1...
6;                      . . . . . 6 1 1 1 1 1...
3, 2, 2;                . . 3 . 2 . 2 1 1 1 1...
5, 2;                   . . . . 5 . 2 1 1 1 1...
4, 3;                   . . . 4 . . 3 1 1 1 1...
7;                      . . . . . . 7 1 1 1 1...

Omar E. Pol, Illustration of the partitions and regions of n, for n = 1..12
Mikhail Kurkov, PARI program. [verification needed]

Column 1 is A141285. Row n has length A194548(n), n > 1.

Cf. A135010, A138121, A182703, A194711, A206437, A210942, A210943.

a210941(n)={
    my(p=[],r=[1]);
    if(n>1,
    my(c=2);
    while(#r1]));
        c++));
    return(r[1..n])
} \\ Joe Slater, Sep 02 2024

cp's OEIS Frontend

A340056 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

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Mathematica

A206555 Number of 5's in the last section of the set of partitions of n.

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Sage

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A206560 Number of 10's in the last section of the set of partitions of n.

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Sage

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A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

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A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

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A134869 Row sums of triangle A134868.

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Maple

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A206562 Triangle read by rows: T(n,k) = sum of all parts >= k in the last section of the set of partitions of n.

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A207379 Triangle read by rows: T(n,k) = number of parts that are in the k-th column of the last section of the set of partitions of n.

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A209655 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

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