A138121 -id:A138121 - cp's OEIS frontend

A194438 Triangle read by rows: T(n,k) is the number of regions of the set of partitions of n into k parts.

1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 3, 1, 1, 0, 1, 0, 1, 5, 2, 1, 0, 1, 0, 1, 0, 0, 0, 1, 7, 3, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 11, 4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 15, 6, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0

Offset: 1

Omar E. Pol, Nov 28 2011

For the definition of "region" see A206437. See also A186114 and A193870. - Omar E. Pol, May 21 2021

			Triangle begins:
   1;
   1,1;
   1,1,1;
   2,1,1,0,1;
   3,1,1,0,1,0,1;
   5,2,1,0,1,0,1,0,0,0,1;
   7,3,1,0,1,0,1,0,0,0,1,0,0,0,1;
  11,4,1,1,1,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1;
...

Omar E. Pol, Illustration of the seven regions of 5

Column 1 is A194439.
Row n has length A000041(n).
Row sums give A000041, n >= 1.
Cf. A008284, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194439, A194446, A194447, A206437.

Definition clarified by Omar E. Pol, May 21 2021

A182181 Total number of parts in the section model of partitions of A135010 with n regions.

Original entry on oeis.org

1, 3, 6, 7, 12, 13, 20, 21, 23, 24, 35, 36, 38, 39, 54, 55, 57, 58, 62, 63, 64, 86, 87, 89, 90, 94, 95, 97, 98, 128, 129, 131, 132, 136, 137, 138, 145, 146, 148, 149, 150, 192, 193, 195, 196, 200, 201, 203, 204, 212, 213, 214, 217, 218, 219, 275

Offset: 1

Omar E. Pol, Apr 23 2012

			The first four regions of the section model of partitions are [1],[2, 1],[3, 1, 1],[2]. We can see that there are seven parts so a(4) = 7.
Written as a triangle begins:
    1;
    3;
    6;
    7,  12;
   13,  20;
   21,  23,  24,  35;
   36,  38,  39,  54;
   55,  57,  58,  62,  63,  64,  86;
   87,  89,  90,  94,  95,  97,  98, 128;
  129, 131, 132, 136, 137, 138, 145, 146, 148, 149, 150, 192;
  193, 195, 196, 200, 201, 203, 204, 212, 213, 214, 217, 218, 219, 275;
  ...
From _Omar E. Pol_, Oct 20 2014: (Start)
Illustration of initial terms:
.                                                _ _ _ _ _
.                                      _ _ _    |_ _ _    |
.                            _ _ _ _  |_ _ _|_  |_ _ _|_  |
.                    _ _    |_ _    | |_ _    | |_ _    | |
.            _ _ _  |_ _|_  |_ _|_  | |_ _|_  | |_ _|_  | |
.      _ _  |_ _  | |_ _  | |_ _  | | |_ _  | | |_ _  | | |
.  _  |_  | |_  | | |_  | | |_  | | | |_  | | | |_  | | | |
. |_| |_|_| |_|_|_| |_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|_|
.
.  1    3      6       7        12        13         20
.
.                                          _ _ _ _ _ _
.                             _ _ _       |_ _ _      |
.                _ _ _ _     |_ _ _|_     |_ _ _|_    |
.   _ _         |_ _    |    |_ _    |    |_ _    |   |
.  |_ _|_ _ _   |_ _|_ _|_   |_ _|_ _|_   |_ _|_ _|_  |
.  |_ _ _    |  |_ _ _    |  |_ _ _    |  |_ _ _    | |
.  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  | |
.  |_ _    | |  |_ _    | |  |_ _    | |  |_ _    | | |
.  |_ _|_  | |  |_ _|_  | |  |_ _|_  | |  |_ _|_  | | |
.  |_ _  | | |  |_ _  | | |  |_ _  | | |  |_ _  | | | |
.  |_  | | | |  |_  | | | |  |_  | | | |  |_  | | | | |
.  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|_|
.
.       21           23           24            35
(End)

Robert Price, Table of n, a(n) for n = 1..204226, rows 1-50.
Omar E. Pol, Illustration of the seven regions of 5

Partial sums of A194446.
Row j has length A187219(j).
Right border gives A006128.
For the definition of "region" see A206437.
Cf. A000041, A135010, A138121, A141285, A182244, A182276, A182727, A186114, A210990.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
reg = {}; l = {};
For[j = 1, j <= 56, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[reg, j - i];
  ];
Accumulate@reg  (* Robert Price, Apr 22 2020, revised Jul 25 2020 *)

a(A000041(n)) = A006128(n), n >= 1.

a(A000041(n)) = A182727(A000041(n)). - Omar E. Pol, May 24 2012

A194437 Triangle read by rows: T(n,k) = sum of parts in the k-th region of n.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 3, 5, 2, 9, 1, 3, 5, 2, 9, 3, 12, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 2, 6, 3, 13, 5, 4, 38, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7

Offset: 1

Omar E. Pol, Nov 27 2011

			Triangle begins:
1;
1,3;
1,3,5;
1,3,5,2,9;
1,3,5,2,9,3,12;
1,3,5,2,9,3,12,2,6,3,20;
1,3,5,2,9,3,12,2,6,3,20,3,7,4,25;
1,3,5,2,9,3,12,2,6,3,20,3,7,4,25,2,6,3,13,5,4,38;
...
Row n has length A000041(n). Row sums give A066186. Right border gives A046746. Records in every row give A046746. Rows converge to A186412.

Cf. A046746, A066186, A135010, A138121, A186114, A186412, A193870, A194436, A194438, A194439, A194446, A194447.

A194805 Number of parts that are visible in one of the three views of the section model of partitions version "tree" with n sections.

Original entry on oeis.org

0, 1, 2, 4, 7, 11, 17, 25, 36, 51, 71, 97, 132, 177, 235, 310, 406, 527, 681, 874, 1116, 1418, 1793, 2256, 2829, 3532, 4393, 5445, 6727, 8282, 10168, 12445, 15190, 18491, 22452, 27192, 32859, 39613, 47651, 57199, 68522, 81920, 97756, 116434, 138435

Offset: 0

Omar E. Pol, Jan 27 2012

nonn

The mentioned view of the section model looks like a tree (see example). Note that every column contains the same parts. For more information about the section model of partitions see A135010 and A194803.

Number of partitions of 2n-1 such that n-1 or n is a part, for n >=1. - Clark Kimberling, Mar 01 2014

			Illustration of one of the three views with seven sections:
.
.                   1
.                 2 1
.                   1 3
.                 2 1
.               4   1
.                   1 3
.                   1   5
.                 2 1
.               4   1
.             3     1
.           6       1
.                     3
.                       5
.                         4
.                           7
.
There are 25 parts that are visible, so a(7) = 25.
Using the formula we have a(7) = p(7) + p(7-1) - 1 = 15 + 11 - 1 = 25, where p(n) is the number of partitions of n.

Robert Price, Table of n, a(n) for n = 0..5000

Cf. A000041, A000065, A084376, A135010, A138121, A141285, A194550, A194803, A194804.

Table[Count[IntegerPartitions[2 n - 1],  p_ /; Or[MemberQ[p, n - 1], MemberQ[p, n]]], {n, 50}]  (* Clark Kimberling, Mar 01 2014 *)
Table[PartitionsP[n] + PartitionsP[n-1] - 1, {n, 0, 44}] (* Robert Price, May 12 2020 *)

a(n) = A084376(n) - 1.
a(n) = A000041(n) + A000041(n-1) - 1, if n >= 1.
a(n) = A000041(n) + A000065(n-1), if n >= 1.

A207380 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

Original entry on oeis.org

0, 3, 10, 21, 42, 70, 122, 187, 298, 443, 667, 957, 1401, 1960, 2775, 3828, 5295, 7167, 9745, 12998, 17380, 22915, 30196, 39347, 51274, 66126, 85209, 108942, 139055, 176273, 223148, 280733, 352623, 440646, 549597, 682411, 845852, 1044084, 1286512, 1579582

Offset: 0

Omar E. Pol, Feb 17 2012

nonn

In this model each part of a partition can be represented by a cuboid of size 1 x 1 x L, where L is the size of the part. One of the views is a rectangle formed by ones whose area is n*A000041(n) = A066186(n). Each element of the first view is equal to the volume of a horizontal column parallel to the axis x. The second view is the n-th slice illustrated in A026792 which has A000041(n) levels and its area is A006128(n) equals the total number of parts of all partitions of n and equals the sum of largest parts of all partitions of n. Each zone contains a partition of n. Each element of the second view is equal to the volume of a horizontal column parallel to the axis y. The third view is a triangle because it is also the n-th slice of the tetrahedron of A209655. The area of triangle is A000217(n). Each element of the third view is equal to the volume of a vertical column parallel to the axis z. The sum of elements of each view is A066186(n) equals the area of the first view. For more information about the shell model of partitions see A135010 and A182703.

			For n = 5 the three views of the three-dimensional shell model of partitions with 5 shells look like this:
.
.   A066186(5) = 35     A006128(5) = 20
.
.         1 1 1 1 1     5
.         1 1 1 1 1     3 2
.         1 1 1 1 1     4 1
.         1 1 1 1 1     2 2 1
.         1 1 1 1 1     3 1 1
.         1 1 1 1 1     2 1 1 1
.         1 1 1 1 1     1 1 1 1 1
.
.
.         7 6 4 2 1
.           1 2 3 2
.             1 1 2
.               1 1
.                 1
.
.   A000217(5) = 15
.
The areas of the shadows of the three views are A066186(5) = 35, A006128(5) = 20 and A000217(5) = 15, therefore the total area of the three shadows is 35+20+15 = 70, so a(5) = 70.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Other versions: A210970, A210979, A210980, A210990, A210991.

Cf. A000041, A000217, A006128, A026792, A066186, A135010, A138121, A141285, A182703, A182715, A206437, A209655.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= n-> n*b(n, n)[1] +b(n, n)[2] +n*(n+1)/2:
seq (a(n), n=0..50);  # Alois P. Heinz, Mar 22 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If [n == 0 || i == 1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; Join[f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]] ]]]; a[n_] := n*b[n, n][[1]] + b[n, n][[2]] + n*(n+1)/2; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 18 2015, after Alois P. Heinz *)

a(n) = n * A000041(n) + A000217(n) + A006128(n) = A066186(n) + A000217(n) + A006128(n).

More terms from Alois P. Heinz, Mar 22 2012

A210970 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

Original entry on oeis.org

0, 3, 9, 18, 34, 55, 91, 136, 208, 301, 439, 616, 876, 1203, 1665, 2256, 3062, 4083, 5459, 7186, 9470, 12335, 16051, 20688, 26648, 34027, 43395, 54966, 69496, 87341, 109591, 136766, 170382, 211293, 261519, 322382, 396694, 486327, 595143, 725954, 883912

Offset: 0

Omar E. Pol, Apr 22 2012

nonn

For more information see A135010 and A182703.

			For n = 6 the illustration of the three views of a three-dimensional version of the shell model of partitions with 6 shells looks like this:
.
.   A006128(6) = 35     A006128(6) = 35
.
.                 6     6
.               3 3     3 3
.               4 2     4 2
.             2 2 2     2 2 2
.               5 1     5 1
.             3 2 1     3 2 1
.             4 1 1     4 1 1
.           2 2 1 1     2 2 1 1
.           3 1 1 1     3 1 1 1
.         2 1 1 1 1     2 1 1 1 1
.       1 1 1 1 1 1     1 1 1 1 1 1
.
.
.       1 2 5 9 12 6  \
.         1 1 3 5 6    \
.           1 1 2 4     \ 6th slice of
.             1 1 2     / tetrahedron A210961
.               1 1    /
.                 1   /
.
.      A000217(6) = 21
.
The areas of the shadows of the three views are A006128(6) = 35, A006128(6) = 35 and A000217(6) = 21, therefore the total area of the three shadows is 35+35+21 = 91, so a(6) = 91.

Other versions: A207380, A210979, A210980, A210990, A210991.

Cf. A000217, A006128, A066186, A135010, A138121, A182703, A206437, A210952, A210961.

a(n) = 2*A006128(n) + A000217(n).

A168020 Square array read by antidiagonals in which row n lists the number of partitions of n into parts divisible by k.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 5, 0, 0, 0, 7, 2, 1, 0, 0, 11, 0, 0, 0, 0, 0, 15, 3, 0, 1, 0, 0, 0, 22, 0, 2, 0, 0, 0, 0, 0, 30, 5, 0, 0, 1, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 11, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Omar E. Pol, Nov 20 2009

In the square array, note that the column k starts with k-1 zeros. Then list each partition number of positive integers followed by k-1 zeros. See A000041, which is the main entry for this sequence.

			The array, A(n, k), begins:
   n | k = 1   2   3   4   5   6   7   8   9  10  11  12
  ---+--------------------------------------------------
   1 |     1   0   0   0   0   0   0   0   0   0   0   0
   2 |     2   1   0   0   0   0   0   0   0   0   0   0
   3 |     3   0   1   0   0   0   0   0   0   0   0   0
   4 |     5   2   0   1   0   0   0   0   0   0   0   0
   5 |     7   0   0   0   1   0   0   0   0   0   0   0
   6 |    11   3   2   0   0   1   0   0   0   0   0   0
   7 |    15   0   0   0   0   0   1   0   0   0   0   0
   8 |    22   5   0   2   0   0   0   1   0   0   0   0
   9 |    30   0   3   0   0   0   0   0   1   0   0   0
  10 |    42   7   0   0   2   0   0   0   0   1   0   0
  11 |    56   0   0   0   0   0   0   0   0   0   1   0
  12 |    77  11   5   3   0   2   0   0   0   0   0   1
  ...
Antidiagonal triangle, T(n,k), begins as:
   1;
   2, 0;
   3, 1, 0;
   5, 0, 0, 0;
   7, 2, 1, 0, 0;
  11, 0, 0, 0, 0, 0;
  15, 3, 0, 1, 0, 0, 0;
  22, 0, 2, 0, 0, 0, 0, 0;
  30, 5, 0, 0, 1, 0, 0, 0, 0;
  42, 0, 0, 0, 0, 0, 0, 0, 0, 0;

G. C. Greubel, Antidiagonals n = 1..50, flattened
Omar E. Pol, Illustration of the shell model of partitions (2D and 3D)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000007, A000041, A035377, A035444, A135010, A138121.

Cf. A168016, A168017, A168018, A168019, A168021. [From Omar E. Pol, Nov 23 2009]

T[n_, k_]:= If[IntegerQ[(n-k+1)/k], PartitionsP[(n-k+1)/k], 0];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)

def A168020(n,k): return number_of_partitions((n-k+1)/k) if ((n-k+1)%k)==0 else 0
flatten([[A168020(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jan 12 2023

A(n, k) = A000041(n/k) if k divides n, otherwise A(n, k) = 0 (array).
A(n, 1) = A(n*k, k) = A000041(n).
From G. C. Greubel, Jan 12 2023: (Start)
T(n, k) = A000041((n-k+1)/k) if k divides (n-k+1), otherwise T(n, k) = 0 (triangle).
T(n, 1) = A000041(n).
T(2*n, n) = 2*A000007(n-1), n >= 1. (End)

Edited by Omar E. Pol, Nov 21 2009
Edited by Charles R Greathouse IV, Mar 23 2010
Edited by Max Alekseyev, May 07 2010

A182244 Sum of all parts of the shell model of partitions of A135010 with n regions.

Original entry on oeis.org

1, 4, 9, 11, 20, 23, 35, 37, 43, 46, 66, 69, 76, 80, 105, 107, 113, 116, 129, 134, 138, 176, 179, 186, 190, 204, 207, 216, 221, 270, 272, 278, 281, 294, 299, 303, 326, 330, 340, 346, 351, 420, 423, 430, 434, 448, 451, 460, 465, 492, 497, 501, 516, 523, 529, 616

Offset: 1

Omar E. Pol, Apr 23 2012

			The first four regions of the shell model of partitions are [1],[2, 1],[3, 1, 1],[2], so a(4) = (1)+(2+1)+(3+1+1)+(2) = 11.
Written as a triangle begins:
1;
4;
9;
11,  20;
23,  35;
37,  43, 46, 66;
69,  76, 80,105;
107,113,116,129,134,138,176;
179,186,190,204,207,216,221,270;
272,278,281,294,299,303,326,330,340,346,351,420;
423,430,434,448,451,460,465,492,497,501,516,523,529,616;
...
From _Omar E. Pol_, Aug 08 2013: (Start)
Illustration of initial terms:
.                                                _ _ _ _ _
.                                      _ _ _    |_ _ _    |
.                            _ _ _ _  |_ _ _|_  |_ _ _|_  |
.                    _ _    |_ _    | |_ _    | |_ _    | |
.            _ _ _  |_ _|_  |_ _|_  | |_ _|_  | |_ _|_  | |
.      _ _  |_ _  | |_ _  | |_ _  | | |_ _  | | |_ _  | | |
.  _  |_  | |_  | | |_  | | |_  | | | |_  | | | |_  | | | |
. |_| |_|_| |_|_|_| |_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|_|
.
.  1    4      9       11       20        23        35
.
.                                          _ _ _ _ _ _
.                             _ _ _       |_ _ _      |
.                _ _ _ _     |_ _ _|_     |_ _ _|_    |
.   _ _         |_ _    |    |_ _    |    |_ _    |   |
.  |_ _|_ _ _   |_ _|_ _|_   |_ _|_ _|_   |_ _|_ _|_  |
.  |_ _ _    |  |_ _ _    |  |_ _ _    |  |_ _ _    | |
.  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  | |
.  |_ _    | |  |_ _    | |  |_ _    | |  |_ _    | | |
.  |_ _|_  | |  |_ _|_  | |  |_ _|_  | |  |_ _|_  | | |
.  |_ _  | | |  |_ _  | | |  |_ _  | | |  |_ _  | | | |
.  |_  | | | |  |_  | | | |  |_  | | | |  |_  | | | | |
.  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|_|
.
.       37           43           46           66
(End)

Omar E. Pol, Illustration of the seven regions of 5

Partial sums of A186412. Row j has length A187219(j). Right border gives A066186.

Cf. A000041, A135010, A138121, A141285, A182244, A182181, A182276, A186114, A206437, A210990.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A186412 = {}; l = {};
For[j = 1, j <= 56, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[A186412, Total@Take[Reverse[First /@ lex[mx]], j - i]];
  ];
Accumulate@A186412  (* Robert Price, Jul 25 2020 *)

a(A000041(k)) = A066186(k), k >= 1.

A182715 Triangle read by rows in which row n lists in nonincreasing order the smallest part of every partition of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Dec 01 2010

Triangle read by rows in which row n lists the smallest parts of all partitions of n in the order produced by the shell model of partitions of A138121.
Also, row n lists the "filler parts" of all partition of n. For more information see A182699.
Row n has length A000041(n). Row sums give A046746. Column 1 gives A001477. The last A000041(n-1) terms of row n are ones, n >= 1.

			For row 10, see the illustration of the link.
Triangle begins:
  0,
  1,
  2,1,
  3,1,1,
  4,2,1,1,1,
  5,2,1,1,1,1,1,
  6,3,2,2,1,1,1,1,1,1,1,
  7,3,2,2,1,1,1,1,1,1,1,1,1,1,1,
  8,4,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  9,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  ...

Omar E. Pol, Illustration of the smallest parts of all partitions of 10 (colored blue) in the shell model of partitions

Cf. A026807, A135010, A138121, A141285, A182699, A182709.

Mirror of triangle A196931.

Name simplified and more terms from Omar E. Pol, Oct 21 2011

A182731 Odd-indexed rows of triangle A141285.

Original entry on oeis.org

1, 3, 3, 5, 3, 5, 4, 7, 3, 5, 4, 7, 3, 6, 5, 9, 3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11

Offset: 1

Omar E. Pol, Nov 28 2010

			Triangle begins:
1,
3,
3, 5,
3, 5, 4, 7,
3, 5, 4, 7, 3, 6, 5, 9,
3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11,

Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A135010, A138121, A141285, A182730, A182732.

Rows converge to A182733.

cp's OEIS Frontend

A194438 Triangle read by rows: T(n,k) is the number of regions of the set of partitions of n into k parts.

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A182181 Total number of parts in the section model of partitions of A135010 with n regions.

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A194437 Triangle read by rows: T(n,k) = sum of parts in the k-th region of n.

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A194805 Number of parts that are visible in one of the three views of the section model of partitions version "tree" with n sections.

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A207380 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

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Maple

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A210970 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

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A168020 Square array read by antidiagonals in which row n lists the number of partitions of n into parts divisible by k.

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A182244 Sum of all parts of the shell model of partitions of A135010 with n regions.

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