A182743 -id:A182743 - cp's OEIS frontend

A182983 Triangle read by rows: row n lists the parts of the n-th shell of the table A182743.

3, 2, 5, 2, 2, 3, 4, 7, 2, 2, 2, 2, 3, 3, 3, 3, 6, 4, 5, 9, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 3, 4, 4, 3, 8, 4, 7, 5, 6, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 3, 3, 7, 3, 4, 6, 3, 5, 5, 3, 10, 4, 4, 5, 4, 9, 5, 8, 6, 7, 13, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Jan 26 2011

Apparently this is the main table for odd numbers of the shell model of partitions. It appears that the table shows an overlapping of all the heads of last sections of partitions of all odd numbers. This is the table 2.1 mentioned in A135010, a geometric version of the table A182743. For even numbers see A182982. The largest parts of the rows of the diagram give A182733.

			Triangle begins:
3,
2, 5,
2, 2, 3, 4, 7,
2, 2, 2, 2, 3, 3, 3, 3, 6, 4, 5, 9

Cf. A135010, A141285, A182733, A182743, A182980, A182982.

A182745 Second column of the table A182743.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 30 2010

nonn

The second largest of the n-th partition of the table A182743.

Cf. A135010, A138121, A141285, A182731, A182733, A182742, A182743, A182744.

A138121 Triangle read by rows in which row n lists the partitions of n that do not contain 1 as a part in juxtaposed reverse-lexicographical order followed by A000041(n-1) 1's.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 21 2008

Mirror of triangle A135010.

			Triangle begins:
[1];
[2],[1];
[3],[1],[1];
[4],[2,2],[1],[1],[1];
[5],[3,2],[1],[1],[1],[1],[1];
[6],[3,3],[4,2],[2,2,2],[1],[1],[1],[1],[1],[1],[1];
[7],[4,3],[5,2],[3,2,2],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1];
...
The illustration of the three views of the section model of partitions (version "tree" with seven sections) shows the connection between several sequences.
---------------------------------------------------------
Partitions                A194805            Table 1.0
.  of 7       p(n)        A194551             A135010
---------------------------------------------------------
7              15                    7     7 . . . . . .
4+3                                4       4 . . . 3 . .
5+2                              5         5 . . . . 2 .
3+2+2                          3           3 . . 2 . 2 .
6+1            11    6       1             6 . . . . . 1
3+3+1                  3     1             3 . . 3 . . 1
4+2+1                    4   1             4 . . . 2 . 1
2+2+2+1                    2 1             2 . 2 . 2 . 1
5+1+1           7            1   5         5 . . . . 1 1
3+2+1+1                      1 3           3 . . 2 . 1 1
4+1+1+1         5        4   1             4 . . . 1 1 1
2+2+1+1+1                  2 1             2 . 2 . 1 1 1
3+1+1+1+1       3            1 3           3 . . 1 1 1 1
2+1+1+1+1+1     2          2 1             2 . 1 1 1 1 1
1+1+1+1+1+1+1   1            1             1 1 1 1 1 1 1
.               1                         ---------------
.               *<------- A000041 -------> 1 1 2 3 5 7 11
.                         A182712 ------->   1 0 2 1 4 3
.                         A182713 ------->     1 0 1 2 2
.                         A182714 ------->       1 0 1 1
.                                                  1 0 1
.                         A141285           A182703  1 0
.                    A182730   A182731                 1
---------------------------------------------------------
.                              A138137 --> 1 2 3 6 9 15..
---------------------------------------------------------
.       A182746 <--- 4 . 2 1 0 1 2 . 4 ---> A182747
---------------------------------------------------------
.
.       A182732 <--- 6 3 4 2 1 3 5 4 7 ---> A182733
.                    . . . . 1 . . . .
.                    . . . 2 1 . . . .
.                    . 3 . . 1 2 . . .
.      Table 2.0     . . 2 2 1 . . 3 .     Table 2.1
.                    . . . . 1 2 2 . .
.                            1 . . . .
.
.  A182982  A182742       A194803       A182983  A182743
.  A182992  A182994       A194804       A182993  A182995
---------------------------------------------------------
.
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms (n = 1..6). The table shows the six sections of the set of partitions of 6. Note that before the dissection the set of partitions was in the ordering mentioned in A026792. More generally, 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.
Illustration of initial terms:
---------------------------------------
n  j     Diagram          Parts
---------------------------------------
.         _
1  1     |_|              1;
.         _ _
2  1     |_  |            2,
2  2       |_|            .  1;
.         _ _ _
3  1     |_ _  |          3,
3  2         | |          .  1,
3  3         |_|          .  .  1;
.         _ _ _ _
4  1     |_ _    |        4,
4  2     |_ _|_  |        2, 2,
4  3           | |        .  1,
4  4           | |        .  .  1,
4  5           |_|        .  .  .  1;
.         _ _ _ _ _
5  1     |_ _ _    |      5,
5  2     |_ _ _|_  |      3, 2,
5  3             | |      .  1,
5  4             | |      .  .  1,
5  5             | |      .  .  1,
5  6             | |      .  .  .  1,
5  7             |_|      .  .  .  .  1;
.         _ _ _ _ _ _
6  1     |_ _ _      |    6,
6  2     |_ _ _|_    |    3, 3,
6  3     |_ _    |   |    4, 2,
6  4     |_ _|_ _|_  |    2, 2, 2,
6  5               | |    .  1,
6  6               | |    .  .  1,
6  7               | |    .  .  1,
6  8               | |    .  .  .  1,
6  9               | |    .  .  .  1,
6  10              | |    .  .  .  .  1,
6  11              |_|    .  .  .  .  .  1;
...
(End)

Robert Price, Table of n, a(n) for n = 1..16851, 25 rows
Omar E. Pol, A section model of partitions (2D and 3D) [From _Omar E. Pol_, Sep 07 2008]
Robert Price, Mathematica Program to Generate Diagram

Row n has length A138137(n).
Rows sums give A138879.
Cf. A000041, A135010, A138879, A138880, A141285, A182703, A194812, A206437, A211009.

less[run1_, run2_] := (lg1 = run1 // Length; lg2 = run2 // Length; lg = Max[lg1, lg2]; r1 = If[lg1 == lg, run1, PadRight[run1, lg, 0]]; r2 = If[lg2 == lg, run2, PadRight[run2, lg, 0]]; Order[r1, r2] != -1); row[n_] := Join[Array[1 &, {PartitionsP[n - 1]}], Sort[Reverse /@ Select[IntegerPartitions[n], FreeQ[#, 1] &], less]] // Flatten // Reverse; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 15 2013 *)
Table[Reverse/@Reverse@DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x]!=1]], x_ /; x==0, 2]~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 1, 9}]  // Flatten (* Robert Price, May 11 2020 *)

A182703 Triangle read by rows: T(n,k) = number of occurrences of k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 2, 0, 1, 5, 1, 1, 0, 1, 7, 4, 2, 1, 0, 1, 11, 3, 2, 1, 1, 0, 1, 15, 8, 3, 3, 1, 1, 0, 1, 22, 7, 6, 2, 2, 1, 1, 0, 1, 30, 15, 6, 5, 3, 2, 1, 1, 0, 1, 42, 15, 10, 5, 4, 2, 2, 1, 1, 0, 1, 56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1

Offset: 1

Omar E. Pol, Nov 28 2010

For the definition of "section" of the set of partitions of n see A135010.
Also, column 1 gives the number of partitions of n-1. For k >= 2, row n lists the number of k's in all partitions of n that do not contain 1 as a part.
From Omar E. Pol, Feb 12 2012: (Start)
It appears that reversed rows converge to A002865.
It appears that row n is also the base of an isosceles triangle in which the column sums give the partition numbers A000041 in descending order starting with p(n-1) = A000041(n-1). Example for n = 7:
.
.         1,
.      1, 0, 1,
.   4, 2, 1, 0, 1,
11, 3, 2, 1, 1, 0, 1,
---------------------
11, 7, 5, 3, 2, 1, 1,
.
It appears that in row n starts an infinite trapezoid in which column sums always give the number of partitions of n-1. Example for n = 7:
.
11, 3, 2, 1, 1, 0, 1,
.   8, 3, 3, 1, 1, 0, 1,
.      6, 2, 2, 1, 1, 0, 1,
.         5, 3, 2, 1, 1, 0, 1,
.            4, 2, 2, 1, 1, 0, 1,
.               5, 2, 2, 1, 1, 0,...
.                  4, 2, 2, 1, 1,...
.                     4, 2, 2, 1,...
.                        4, 2, 2,...
.                           4, 2,...
.                              4,...
.
The sum of any column is always p(7-1) = p(6) = A000041(6) = 11.
It appears that the first term of row n is one of the vertices of an infinite isosceles triangle in which column sums give the partition numbers A000041 in ascending order starting with p(n-1) = A000041(n-1). Example for n = 7:
11,
.    8,
.    7,  6,
.        6,  5,
.       10,  5, ...
.           10, ...
.           10, ...
-------------------
11, 15, 22, 30, ...
(End)
It appears that row n lists the first differences of the row n of triangle A207031 together with 1 (as the final term of row n). - Omar E. Pol, Feb 26 2012
More generally T(n,k) is the number of occurrences of k in the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Oct 21 2013

			Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7:
.                                        _ _ _ _ _ _ _
.     (7)                    (7)        |_ _ _ _      |
.     (4+3)                (4+3)        |_ _ _ _|_    |
.     (5+2)                (5+2)        |_ _ _    |   |
.     (3+2+2)            (3+2+2)        |_ _ _|_ _|_  |
.       (1)                  (1)                    | |
.         (1)                (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.           (1)              (1)                    | |
.             (1)            (1)                    | |
.             (1)            (1)                    | |
.               (1)          (1)                    | |
.                 (1)        (1)                    |_|
.    ----------------
.     19,8,5,3,2,1,1 --> Row 7 of triangle A207031.
.      |/|/|/|/|/|/|
.     11,3,2,1,1,0,1 --> Row 7 of this triangle.
.
Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So, for k = 1..7, row 7 gives: 11, 3, 2, 1, 1, 0, 1.
Triangle begins:
   1;
   1,  1;
   2,  0,  1;
   3,  2,  0,  1;
   5,  1,  1,  0, 1;
   7,  4,  2,  1, 0, 1;
  11,  3,  2,  1, 1, 0, 1;
  15,  8,  3,  3, 1, 1, 0, 1;
  22,  7,  6,  2, 2, 1, 1, 0, 1;
  30, 15,  6,  5, 3, 2, 1, 1, 0, 1;
  42, 15, 10,  5, 4, 2, 2, 1, 1, 0, 1;
  56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Omar E. Pol, Illustration of the section model of partitions, Figure 1 (n = 1..6), Figure 2 (2D view, n = 1..10), Figure 3 (3D view, n = 1..9)

Row sums give A138137. Where records occur is A134869.
Columns 1-10: A000041, A182712-A182714, A206555-A206560.
Sub-triangles (1-11): A023531, A129186, A194702-A194710
Cf. A066633, A135010, A182742, A182743, A194812, A206563, A207031, A207032, A206437, A211025.

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n,i) option remember; local g;
      if n=0        then [1]
    elif n<2 or i<2 then [0]
    else g:=   `if`(i>n, [0],  b(n-i, i));
         p(p([0$j=2..i, g[1]], b(n, i-1)), g)
      fi
    end:
h:= proc(n) option remember;
      `if`(n=0, 1, b(n, n)[1]+h(n-1))
    end:
T:= proc(n) h(n-1), b(n, n)[2..n][] end:
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 19 2012

p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{g}, Which[n == 0, {1}, n<2 || i<2, {0}, True, g = If [i>n, {0}, b[n-i, i]]; p[p[Append[Array[0&, i-1], g[[1]]], b[n, i-1]], g]]]; h[n_] := h[n] = If[n == 0, 1, b[n, n][[1]] + h[n-1]]; t[n_] := {h[n-1], Sequence @@ b[n, n][[2 ;; n]]}; Table[t[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 16 2014, after Alois P. Heinz's Maple code *)
Table[{PartitionsP[n-1]}~Join~Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], k], {k,2,n}], {n,1,12}]  // Flatten (* Robert Price, May 15 2020 *)

It appears that T(n,k) = A207032(n,k) - A207032(n,k+2). - Omar E. Pol, Feb 26 2012

A182742 Table of partitions that do not contain 1 as a part for even integers.

Original entry on oeis.org

2, 4, 2, 3, 2, 2, 6, 3, 2, 2, 5, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 8, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 7, 3, 2, 2, 2, 2, 2, 2, 2, 6, 3, 3, 2, 2, 2, 2, 2, 2, 2, 5, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 3, 2, 2

Offset: 1

Omar E. Pol, Nov 30 2010, Dec 01 2010, Dec 04 2010

This array read by antidiagonals is the main table of the shell model of partitions for even integers. Here the last sections of all even numbers are superimposed as shells of an onion. In this way many bits of information are saved.
The table is the head of the last section of partitions of an even integer when it tends to be infinite. Row n lists the parts of the n-th partition that do not contains 1 as a part.
The shell model of partitions uses this table during the filling mechanism of the head of the last section of the next even integer k. For example, in a mechanical version, the head of the last section (as a mirror) pivoting from vertical to horizontal position. Then a copy of the partitions of the integer k, listed in this table, is transmitted (or reflected) at the head (or mirror) of the last section. Finally the head (or mirror) pivots back to return to its original vertical position. And so on for all even integers.
In another version, simply a copy of the partitions of the integer k, listed in the table, are placed above the partitions of the last odd number placed in the vertical plane structure.
It appears this table is useful to know the structure of the partitions of all even integers. The same applies for odd numbers in the table of A182743. Furthermore, both tables can be unified in a three-dimensional shell model.

			Array begins:
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
3, 3, 2, 2, 2, 2, 2, 2, 2, 2,
6, 2, 2, 2, 2, 2, 2, 2, 2,
5, 3, 2, 2, 2, 2, 2, 2,
4, 4, 2, 2, 2, 2, 2,
8, 2, 2, 2, 2, 2,
4, 3, 3, 2, 2,
7, 3, 2, 2,
6, 4, 2,
5, 5,
10,

Cf. A135010, A138121, A141285, A182730, A182736, A182740, A182743, A182746.

Column 1 give A182732. Column 2 give A182744.

A182709 Sum of the emergent parts of the partitions of n.

Original entry on oeis.org

0, 0, 0, 2, 3, 11, 14, 33, 45, 81, 109, 185, 237, 372, 490, 715, 928, 1326, 1693, 2348, 2998, 4032, 5119, 6795, 8530, 11132, 13952, 17927, 22314, 28417, 35126, 44279, 54532, 68062, 83422, 103427, 126063, 155207, 188506, 230547, 278788, 339223, 408482

Offset: 1

Omar E. Pol, Nov 28 2010, Nov 29 2010

Here the "emergent parts" of the partitions of n are defined to be the parts (with multiplicity) of all the partitions that do not contain "1" as a part, removed by one copy of the smallest part of every partition. Note that these parts are located in the head of the last section of the set of partitions of n. For more information see A182699.

Also total sum of parts of the regions that do not contain 1 as a part in the last section of the set of partitions of n (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012

			For n=7 the partitions of 7 that do not contain "1" as a part are
7
4 + 3
5 + 2
3 + 2 + 2
Then remove one copy of the smallest part of every partition. The rest are the emergent parts:
.,
4, .
5, .
3, 2, .
The sum of these parts is 4 + 5 + 3 + 2 = 14, so a(7)=14.
For n=10 the illustration in the link shows the location of the emergent parts (colored yellow and green) and the location of the filler parts (colored blue) in the last section of the set of partitions of 10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jason Kimberley)
Omar E. Pol, Illustration: How to build the last section of the set of partitions (copy, paste and fill)
Omar E. Pol, Illustration of the shell model of partitions (2D view)

Cf. A000041, A135010, A138121, A138879, A138880, A182699, A182703, A182708, A182740, A182742, A182743.

Row sums of A183152.

b:= proc(n, i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
c:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then k
    elif i<2 then 0
    else c(n, i-1, k) +c(n-i, i, i)
      fi
    end:
a:= n-> n*b(n, n) - c(n, n, 0):
seq(a(n), n=1..40);  #  Alois P. Heinz, Dec 01 2010

f[n_]:=Total[Flatten[Most/@Select[IntegerPartitions[n],!MemberQ[#,1]&]]]; Table[f[i],{i,50}] (* Harvey P. Dale, Dec 28 2010 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n - i, i]]; c[n_, i_, k_] := c[n, i, k] = Which[n<0, 0, n==0, k, i<2, 0, True, c[n, i-1, k] + c[n-i, i, i]]; a[n_] := n*b[n, n] - c[n, n, 0]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Oct 08 2015, after Alois P. Heinz *)

a(n) = A138880(n) - A182708(n).
a(n) = A066186(n) - A066186(n-1) - A046746(n) = A138879(n) - A046746(n). - Omar E. Pol, Aug 01 2013
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (12*sqrt(2*n)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n)). - Vaclav Kotesovec, Jan 03 2019, extended Jul 05 2019

More terms from Alois P. Heinz, Dec 01 2010

A182746 Bisection (even part) of number of partitions that do not contain 1 as a part A002865.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 55, 88, 137, 210, 320, 478, 708, 1039, 1507, 2167, 3094, 4378, 6153, 8591, 11914, 16424, 22519, 30701, 41646, 56224, 75547, 101066, 134647, 178651, 236131, 310962, 408046, 533623, 695578, 903811, 1170827, 1512301, 1947826, 2501928

Offset: 0

Omar E. Pol, Dec 01 2010

a(n+1) is the number of partitions p of 2n-1 such that (number of parts of p) is a part of p, for n >=0. - Clark Kimberling, Mar 02 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marco Baggio, Vasilis Niarchos, Kyriakos Papadodimas, and Gideon Vos, Large-N correlation functions in N = 2 superconformal QCD, arXiv preprint arXiv:1610.07612 [hep-th], 2016.
K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.

Cf. A000041, A002865, A058696, A135010, A138121, A182740, A182742, A182743, A182747.

b:= proc(n, i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
a:= n-> b(2*n, 2*n):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 01 2010

Table[Count[IntegerPartitions[2 n -1], p_ /; MemberQ[p, Length[p]]], {n, 20}]   (* Clark Kimberling, Mar 02 2014 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n-i, i]]; a[n_] := b[2*n, 2*n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
a[n_] := PartitionsP[2*n] - PartitionsP[2*n - 1]; Table[a[n], {n, 0, 40}] (* George Beck, Jun 05 2017 *)

a(n)=numbpart(2*n)-numbpart(2*n-1) \\ Charles R Greathouse IV, Jun 06 2017

a(n) = p(2*n) - p(2*n-1), where p is the partition function, A000041. - George Beck, Jun 05 2017 [Shifted by Georg Fischer, Jun 20 2022]

More terms from Alois P. Heinz, Dec 01 2010

A182747 Bisection (odd part) of number of partitions that do not contain 1 as a part A002865.

Original entry on oeis.org

0, 1, 2, 4, 8, 14, 24, 41, 66, 105, 165, 253, 383, 574, 847, 1238, 1794, 2573, 3660, 5170, 7245, 10087, 13959, 19196, 26252, 35717, 48342, 65121, 87331, 116600, 155038, 205343, 270928, 356169, 466610, 609237, 792906, 1028764, 1330772, 1716486, 2207851

Offset: 0

Omar E. Pol, Dec 01 2010

a(n+1) = number of partitions p of 2n such that (number of parts of p) is a part of p, for n >=0. - Clark Kimberling, Mar 02 2014

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

Cf. A000041, A002865, A058695, A135010, A138121, A182740, A182742, A182743, A182746.

b:= proc(n,i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
a:= n-> b(2*n+1, 2*n+1):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 01 2010

f[n_] := Table[PartitionsP[2 k + 1] - PartitionsP[2 k], {k, 0, n}] (* George Beck, Aug 14 2011 *)
(* also *)
Table[Count[IntegerPartitions[2 n], p_ /; MemberQ[p, Length[p]]], {n, 20}] (* Clark Kimberling, Mar 02 2014 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n == 0, 1, i<2, 0, True, b[n, i-1] + b[n-i, i]]; a[n_] := b[2*n+1, 2*n+1]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) = p(2*n+1)-p(2*n), where p is the partition function, A000041. - George Beck, Aug 14 2011

More terms from Alois P. Heinz, Dec 01 2010

A182995 Sum of parts of the n-th subsection of the head of the last section of the set of partitions of any odd integer >= 2n+1.

Original entry on oeis.org

3, 7, 18, 44, 82, 158, 303, 507, 873, 1470, 2354, 3756, 5923, 9065, 13815, 20824, 30853, 45365, 66210, 95415, 136696, 194414, 274057, 384136, 535219, 740559, 1019529, 1396212, 1901533, 2577918, 3479291, 4673711, 6253003, 8332767

Offset: 1

Omar E. Pol, Feb 06 2011

nonn

The last section of the set of partitions of 2n+1 contains n subsections.

Also first differences of A182737. - Omar E. Pol, Mar 03 2011

			a(5)=82 because the 5th subsection of the head of the last section of any odd integer >= 11 looks like this:
(11 . . . . . . . . . . )
( 6 . . . . . 5 . . . . )
( 7 . . . . . . 4 . . . )
( 8 . . . . . . . 3 . . )
( 4 . . . 4 . . . 3 . . )
( 5 . . . . 3 . . 3 . . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
There are 21 parts whose sum is 11+6+5+7+4+8+3+4+4+3+5+3+3+2+2+2+2+2+2+2+2 = 11*6 + 2*8 = 82, so a(5) = 82.

Cf. A135010, A138880, A182737, A182743, A182983, A182993, A182994.

a(n) = A138880(2n+1) - A138880(2n-1).

a(17) corrected and more terms from Omar E. Pol, Mar 03 2011.

a(12) corrected by Georg Fischer, Aug 31 2020

A182740 A shell model of partitions as a table of partitions.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 2, 0, 1, 1, 4, 0, 0, 1, 1

Offset: 1

Omar E. Pol, Nov 30 2010

This array read by antidiagonals is a table of partitions for all integers.
For another version but as a binary code see A182741.
For more information see A135010 and A138121 which are the main entries for this sequence.

			For the numbers 1..6 the shell model of partitions has 6 shells. The model as a table looks like this:
1 1 1 1 1 1
2 . 1 1 1 1
3 . . 1 1 1
2 . 2 . 1 1
4 . . . 1 1
3 . . 2 . 1
5 . . . . 1
2 . 2 . 2 .
4 . . . 2 .
3 . . 3 . .
6 . . . . .
Then replace the dots by zeros.
Remarks: one number by column, for example 23 is located only in a column, not in two columns.
The table looks like this:
1 1 1 1 1 1
2 0 1 1 1 1
3 0 0 1 1 1
2 0 2 0 1 1
4 0 0 0 1 1
3 0 0 2 0 1
5 0 0 0 0 1
2 0 2 0 2 0
4 0 0 0 2 0
3 0 0 3 0 0
6 0 0 0 0 0
Array begins:
1,1,1,1,1,1,
2,0,1,1,1,
3,0,0,1,
2,0,2,
4,0,
3,

Omar E. Pol, Illustration of the shell model of partitions for 1..6 (2D-3D view)
Omar E. Pol, Illustration of the shell model of partitions for 1..10 (2D view)
Omar E. Pol Illustration of the shell model of partitions for 1..9 (3D view)

Cf. A135010, A138121, A182741, A182742, A182743.

cp's OEIS Frontend

A182983 Triangle read by rows: row n lists the parts of the n-th shell of the table A182743.

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A182745 Second column of the table A182743.

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A138121 Triangle read by rows in which row n lists the partitions of n that do not contain 1 as a part in juxtaposed reverse-lexicographical order followed by A000041(n-1) 1's.

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A182703 Triangle read by rows: T(n,k) = number of occurrences of k in the last section of the set of partitions of n.

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A182742 Table of partitions that do not contain 1 as a part for even integers.

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A182709 Sum of the emergent parts of the partitions of n.

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A182746 Bisection (even part) of number of partitions that do not contain 1 as a part A002865.

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A182747 Bisection (odd part) of number of partitions that do not contain 1 as a part A002865.

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