A138879 -id:A138879 - cp's OEIS frontend

A210953 Triangle read by rows: T(n,k) = sum of all parts in the k-th column of the shell model of partitions considering only the n-th shell and with its parts aligned to the right margin.

1, 0, 3, 0, 0, 5, 0, 0, 2, 9, 0, 0, 0, 3, 12, 0, 0, 0, 2, 9, 20, 0, 0, 0, 0, 3, 11, 25, 0, 0, 0, 0, 2, 9, 22, 38, 0, 0, 0, 0, 0, 3, 14, 28, 49, 0, 0, 0, 0, 0, 2, 9, 26, 44, 69, 0, 0, 0, 0, 0, 0, 3, 14, 37, 55, 87, 0, 0, 0, 0, 0, 0, 2, 9, 29, 62, 83, 123

Offset: 1

Omar E. Pol, Apr 22 2012

			For n = 6 and k = 1..6 the 6th shell looks like this:
-------------------------
k: 1,  2,  3,  4,  5,  6
-------------------------
.                      6
.                  3 + 3
.                  4 + 2
.              2 + 2 + 2
.                      1
.                      1
.                      1
.                      1
.                      1
.                      1
.                      1
.
The sums of all parts in columns 1-6 are
.  0,  0,  0,  2,  9, 20, the same as the 6th row of triangle.
Triangle begins:
1;
0, 3;
0, 0, 5;
0, 0, 2, 9;
0, 0, 0, 3, 12;
0, 0, 0, 2,  9, 20;
0, 0, 0, 0,  3, 11, 25;
0, 0, 0, 0,  2,  9, 22, 38;
0, 0, 0, 0,  0,  3, 14, 28, 49;
0, 0, 0, 0,  0,  2,  9, 26, 44, 69;
0, 0, 0, 0,  0,  0,  3, 14, 37, 55, 87;
0, 0, 0, 0,  0,  0,  2,  9, 29, 62, 83, 123;

Row sums give A138879. Column sums converge to A014153. Right border gives A046746, n >= 1.

Cf. A135010, A138121, A210950, A210951, A210952.

A210956 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, 1, 3, 2, 2, 5, 3, 7, 7, 11, 5, 7, 10, 10, 15, 7, 15, 21, 25, 25, 31, 11, 17, 23, 27, 32, 32, 39, 15, 31, 40, 52, 57, 63, 63, 71, 22, 36, 54, 62, 72, 78, 85, 85, 94, 30, 60, 78, 98, 113, 125, 132, 140, 140, 150, 42, 72, 102, 122, 142, 154, 168, 176, 185, 185, 196

Offset: 1

Omar E. Pol, May 01 2012

Row n lists the partial sums of row n of triangle A207383.

			Triangle begins:
1;
1,   3;
2,   2, 5;
3,   7, 7, 11;
5,   7, 10, 10, 15;
7,  15, 21, 25, 25, 31;
11, 17, 23, 27, 32, 32, 39;
15, 31, 40, 52, 57, 63, 63, 71;
22, 36, 54, 62, 72, 78, 85, 85, 94;

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A000041. Right border gives A138879.

Cf. A135010, A138121, A182703, A206437, A206562, A207031, A207032, 207383, 208476, A210948, A210955.

Row(n)={my(v=vector(n)); v[1]=numbpart(n-1); if(n>1, forpart(p=n, for(k=1, #p, v[p[k]]++), [2,n])); for(k=2, n, v[k]=v[k-1]+k*v[k]); v}
{ for(n=1, 10, print(Row(n))) }

T(n,k) = Sum_{j=1..k} A207383(n,j).

Terms a(46) and beyond from Andrew Howroyd, Feb 19 2020

A211993 A list of ordered partitions of the positive integers.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 18 2012

The order of the partitions of the odd integers is the same as A211992. The order of the partitions of the even integers is the same as A026792.

			A table of partitions.
--------------------------------------------
.              Expanded       Geometric
Partitions     arrangement    model
--------------------------------------------
1;             1;             |*|
--------------------------------------------
2;             . 2;           |* *|
1,1;           1,1;           |o|*|
--------------------------------------------
1,1,1;         1,1,1;         |o|o|*|
2,1;           . 2,1;         |o o|*|
3;             . . 3;         |* * *|
--------------------------------------------
4,;            . . . 4;       |* * * *|
2,2;           . 2,. 2;       |* *|* *|
3,1;           . . 3,1;       |o o o|*|
2,1,1,;        . 2,1,1;       |o o|o|*|
1,1,1,1;       1,1,1,1;       |o|o|o|*|
--------------------------------------------
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|
3,1,1;         . . 3,1,1;     |o o o|o|*|
2,2,1;         . 2,. 2,1;     |o o|o o|*|
4,1;           . . . 4,1;     |o o o o|*|
3,2;           . . 3,. 2;     |* * *|* *|
5;             . . . . 5;     |* * * * *|
--------------------------------------------
6;             . . . . . 6;   |* * * * * *|
3,3;           . . 3,. . 3;   |* * *|* * *|
4,2;           . . . 4,. 2;   |* * * *|* *|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
5,1;           . . . . 5,1;   |o o o o o|*|
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|
4,1,1;         . . . 4,1,1;   |o o o o|o|*|
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Rows sums give A036042, n>=1.
Other versions are A026792, A211992, A211994. See also A211983, A211984, A211989, A211999. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A212001 Triangle read by rows: T(n,k) = sum of all parts of the last n-k+1 shells of n.

Original entry on oeis.org

1, 4, 3, 9, 8, 5, 20, 19, 16, 11, 35, 34, 31, 26, 15, 66, 65, 62, 57, 46, 31, 105, 104, 101, 96, 85, 70, 39, 176, 175, 172, 167, 156, 141, 110, 71, 270, 269, 266, 261, 250, 235, 204, 165, 94, 420, 419, 416, 411, 400, 385, 354, 315, 244, 150, 616, 615

Offset: 1

Omar E. Pol, Apr 26 2012

The set of partitions of n contains n shells (see A135010). It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.

			For n = 5 the illustration shows five sets containing the last n-k+1 shells of 5 and below the sum of all parts of each set:
--------------------------------------------------------
.  S{1-5}     S{2-5}     S{3-5}     S{4-5}     S{5}
--------------------------------------------------------
.  The        Last       Last       Last       The
.  five       four       three      two        last
.  shells     shells     shells     shells     shell
.  of 5       of 5       of 5       of 5       of 5
--------------------------------------------------------
.
.  5          5          5          5          5
.  3+2        3+2        3+2        3+2        3+2
.  4+1        4+1        4+1        4+1          1
.  2+2+1      2+2+1      2+2+1      2+2+1          1
.  3+1+1      3+1+1      3+1+1        1+1          1
.  2+1+1+1    2+1+1+1      1+1+1        1+1          1
.  1+1+1+1+1    1+1+1+1      1+1+1        1+1          1
. ---------- ---------- ---------- ---------- ----------
.     35         34         31         26         15
.
So row 5 lists 35, 34, 31, 26, 15.
.
Triangle begins:
1;
4,     3;
9,     8,   5;
20,   19,  16,  11;
35,   34,  31,  26,  15;
66,   65,  62,  57,  46,  31;
105, 104, 101,  96,  85,  70,  39;
176, 175, 172, 167, 156, 141, 110,  71;
270, 269, 266, 261, 250, 235, 204, 165,  94;
420, 419, 416, 411, 400, 385, 354, 315, 244, 150;

Mirror of triangle A212011. Column 1 is A066186. Right border is A138879.

Cf. A135010, A138121, A182703, A206563, A211980, A212000, A212010.

T(n,k) = A066186(n) - A066186(k-1).

T(n,k) = Sum_{j=k..n} A138879(j).

A212011 Triangle read by rows: T(n,k) = sum of all parts of the last k shells of n.

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 11, 16, 19, 20, 15, 26, 31, 34, 35, 31, 46, 57, 62, 65, 66, 39, 70, 85, 96, 101, 104, 105, 71, 110, 141, 156, 167, 172, 175, 176, 94, 165, 204, 235, 250, 261, 266, 269, 270, 150, 244, 315, 354, 385, 400, 411, 416, 419, 420, 196, 346

Offset: 1

Omar E. Pol, Apr 26 2012

The set of partitions of n contains n shells (see A135010). It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.

			For n = 5 the illustration shows five sets containing the last k shells of 5 and below we can see that the sum of all parts of in each set:
--------------------------------------------------------
.  S{5}       S{4-5}     S{3-5}     S{2-5}     S{1-5}
--------------------------------------------------------
.  The        Last       Last       Last       The
.  last       two        three      four       five
.  shell      shells     shells     shells     shells
.  of 5       of 5       of 5       of 5       of 5
--------------------------------------------------------
.
.  5          5          5          5          5
.  3+2        3+2        3+2        3+2        3+2
.    1        4+1        4+1        4+1        4+1
.      1      2+2+1      2+2+1      2+2+1      2+2+1
.      1        1+1      3+1+1      3+1+1      3+1+1
.        1        1+1      1+1+1    2+1+1+1    2+1+1+1
.          1        1+1      1+1+1    1+1+1+1  1+1+1+1+1
. ---------- ---------- ---------- ---------- ----------
.     15         26         31         34         35
.
So row 5 lists 15, 26, 31, 34, 35.
.
Triangle begins:
1;
3,     4;
5,     8,   9;
11,   16,  19,  20;
15,   26,  31,  34,  35;
31,   46,  57,  62,  65,  66;
39,   70,  85,  96, 101, 104, 105;
71,  110, 141, 156, 167, 172, 175, 176;
94,  165, 204, 235, 250, 261, 266, 269, 270;
150, 244, 315, 354, 385, 400, 411, 416, 419, 420;

Mirror of triangle A212001. Column 1 is A138879. Right border is A066186.

Cf. A135010, A182703, A212000, A212010.

T(n,k) = A066186(n) - A066186(n-k).

T(n,k) = Sum_{j=n-k+1..n} A138879(j).

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 *)

A230440 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of partitions of n that do not contain 1 as a part in colexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 18 2013

The n-th row of triangle lists the parts of the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010.

			Illustration of initial terms (row = 1..6). The table shows the six sections of the set of partitions of 6 in three ways. Note that before the dissection, the set of partitions was in colexicographic order, see A211992. More generally, in a master model, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
---------------------------------------------------------
n  j     Diagram          Parts              Parts
---------------------------------------------------------
.         _
1  1     |_|              1;                 1;
.           _
2  1      _| |              1,                 1,
2  2     |_ _|              2;               2;
.             _
3  1         | |              1,                 1,
3  2      _ _| |              1,               1,
3  3     |_ _ _|              3;             3;
.               _
4  1           | |              1,                 1,
4  2           | |              1,               1,
4  3      _ _ _| |              1,             1,
4  4     |_ _|   |            2,2,           2,2,
4  5     |_ _ _ _|              4;           4;
.                 _
5  1             | |              1,                 1,
5  2             | |              1,               1,
5  3             | |              1,             1,
5  4             | |              1,             1,
5  5      _ _ _ _| |              1,           1,
5  6     |_ _ _|   |            3,2,         3,2,
5  7     |_ _ _ _ _|              5;         5;
.                   _
6  1               | |              1,                 1,
6  2               | |              1,               1,
6  3               | |              1,             1,
6  4               | |              1,             1,
6  5               | |              1,           1,
6  6               | |              1,           1,
6  7      _ _ _ _ _| |              1,         1,
6  8     |_ _|   |   |          2,2,2,       2,2,2,
6  9     |_ _ _ _|   |            4,2,       4,2,
6  10    |_ _ _|     |            3,3,       3,3,
6  11    |_ _ _ _ _ _|              6;       6;
...
Triangle begins:
[1];
[1],[2];
[1],[1],[3];
[1],[1],[1],[2,2],[4];
[1],[1],[1],[1],[1],[3,2],[5];
[1],[1],[1],[1],[1],[1],[1],[2,2,2],[4,2],[3,3],[6];
...

Positive terms of A228716.
Row n has length A138137(n).
Row sums give A138879.
Right border gives A000027.
Cf. A000041, A135010, A138121, A141285, A182703, A187219, A193870, A194446, A206437, A207031, A207034, A207383, A207379, A211009.

A207377 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nondecreasing order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 23 2012

Starting from the first row; it appears that the total numbers of occurrences of k in k successive rows give the sequence A000041. For more information see A182703.

			Written as a triangle:
1;
1,2;
1,1,3;
1,1,1,2,2,4;
1,1,1,1,1,2,3,5;
1,1,1,1,1,1,1,2,2,2,2,3,3,4,6;
1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,5,7;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,5,6,8;

Alois P. Heinz, Rows n = 1..23, flattened

Triangle similar to A135010. Mirror of A207378. Row n has length A138137(n). Row sums give A138879. Right border is A000027.

Cf. A000041, A006128, A066186, A138121, A182703, A182712-A182714, A194812.

A208474 Sum of the sizes of the Durfee squares of all partitions of n that do not contain 1 as a part, but with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 7, 13, 16, 24, 30, 46, 55, 79, 100, 136, 169, 229, 282, 374, 462, 598, 737, 947, 1158, 1466, 1794, 2246, 2733, 3399, 4116, 5076, 6133, 7503, 9033, 10993, 13177, 15943, 19061, 22939, 27327, 32749, 38883, 46395, 54938, 65278, 77070, 91270

Offset: 1

Omar E. Pol, Mar 03 2012

nonn

Also sum of the sizes of the Durfee squares of all partitions of the head of the last section of n (see A135010).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews, Partitions and Durfee Dissection
Eric Weisstein's World of Mathematics, Durfee Square

First differences of A115995.

Cf. A135010, A138135, A138137, A138879.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
g:= proc(n) option remember;
      add(add(b(k, d)*b(n-d^2-k, d),
              k=0..n-d^2)*d, d=1..floor(sqrt(n)))
    end:
a:= n-> g(n)-g(n-1):
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 09 2012

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; g[n_] := Sum[Sum[b[k, d]*b[n-d^2-k, d], {k, 0, n-d^2}]*d, {d, 1, Sqrt[n]}]; Table[g[n], {n, 0, 70}] // Differences (* Jean-François Alcover, Feb 21 2017, after Alois P. Heinz *)

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 03 2019

A211025 Triangle read by rows: T(n,k) = total sum of parts in the last section of the set of partitions of n after k-th stage.

Original entry on oeis.org

1, 1, 3, 1, 2, 5, 1, 2, 3, 5, 7, 11, 1, 2, 3, 4, 5, 7, 10, 15, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 19, 22, 25, 31, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 18, 20, 25, 28, 32, 39, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 23, 25, 27

Offset: 1

Omar E. Pol, Apr 25 2012

Also triangle read by rows in which row n lists the partial sums of row n of triangle A135010.

This triangle shows the growth of the last sections of the partitions of n step by step. At stage k one part of size A135010(n,k) is added to the structure of the n-th shell.

			For row n = 5 of triangle we have:
-------------------------------------
Column  Zone    The 5th    Total sum
k                shell     of parts
-------------------------------------
8    <>   7       (5)         15
7    <>   6    (3...          10
6    =    6     ...2)          7
5    =    5       (1)          5
4    =    4       (1)          4
3    =    3       (1)          3
2    =    2       (1)          2
1    =    1       (1)          1
.
Triangle begins:
1;
1,3;
1,2,5;
1,2,3,5,7,11;
1,2,3,4,5,7,10,15;
1,2,3,4,5,6,7,9,11,13,15,19,22,25,31;
1,2,3,4,5,6,7,8,9,10,11,13,15,18,20,25,28,32,39;

Row n has length A138137(n). Right border gives A138879.

Cf. A135010, A182703, A206437, A211030.

cp's OEIS Frontend

A210953 Triangle read by rows: T(n,k) = sum of all parts in the k-th column of the shell model of partitions considering only the n-th shell and with its parts aligned to the right margin.

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A210956 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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PARI

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A211993 A list of ordered partitions of the positive integers.

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A212001 Triangle read by rows: T(n,k) = sum of all parts of the last n-k+1 shells of n.

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A212011 Triangle read by rows: T(n,k) = sum of all parts of the last k shells of 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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Mathematica

A230440 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of partitions of n that do not contain 1 as a part in colexicographic order.

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A207377 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nondecreasing order.

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A208474 Sum of the sizes of the Durfee squares of all partitions of n that do not contain 1 as a part, but with a(1) = 1.

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Maple

Mathematica

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A211025 Triangle read by rows: T(n,k) = total sum of parts in the last section of the set of partitions of n after k-th stage.

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