A194812 -id:A194812 - cp's OEIS frontend

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.

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

A182712 Number of 2's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409

Offset: 0

Omar E. Pol, Nov 28 2010

Essentially the same as A087787 but here a(n) is the number of 2's in the last section of n, not n-2. See also A100818.
Note that a(1)..a(11) coincide with a(2)..a(12) of A005291.
Also number of 2's in all partitions of n that do not contain 1's as a part, if n >= 1. Also 0 together with the first differences of A024786. - Omar E. Pol, Nov 13 2011
Also number of 2's in the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010. - Omar E. Pol, Dec 01 2013

			a(6) = 4 counts the 2's in 6 = 4+2 = 2+2+2. The 2's in 6 = 3+2+1 = 2+2+1+1 = 2+1+1+1+1 do not count. - _Omar E. Pol_, Nov 13 2011
From _Omar E. Pol_, Oct 27 2012: (Start)
----------------------------------
Last section               Number
of the set of                of
partitions of 6             2's
----------------------------------
6 .......................... 0
3 + 3 ...................... 0
4 + 2 ...................... 1
2 + 2 + 2 .................. 3
.   1 ...................... 0
.       1 .................. 0
.       1 .................. 0
.           1 .............. 0
.           1 .............. 0
.               1 .......... 0
.                   1 ...... 0
---------------------------------
.   8 - 4 =                  4
.
In the last section of the set of partitions of 6 the difference between the sum of the second column and the sum of the third column is 8 - 4 = 4, the same as the number of 2's, so a(6) = 4 (see also A024786).
(End)

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

Column 2 of A194812.

Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182713, A182714.

Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 2], {n, 0, 49}] (* Robert Price, May 15 2020 *)

A182712 = lambda n: sum(list(p).count(2) for p in Partitions(n) if 1 not in p) # Omar E. Pol, Nov 13 2011

It appears that A000041(n) = a(n+1) + a(n+2), n >= 0. - Omar E. Pol, Feb 04 2012
G.f.: (x^2/(1 + x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jun 02 2018

A182714 Number of 4's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 3, 2, 5, 5, 10, 10, 17, 19, 31, 34, 51, 60, 86, 100, 139, 165, 223, 265, 349, 418, 543, 648, 827, 992, 1251, 1495, 1866, 2230, 2758, 3289, 4033, 4803, 5852, 6949, 8411, 9973, 12005, 14194, 17002, 20060, 23919, 28153, 33426, 39256, 46438

Offset: 1

Omar E. Pol, Nov 13 2011

nonn

Zero together with the first differences of A024788.
Also number of 4's in all partitions of n that do not contain 1 as a part.
a(n) is the number of partitions of n such that m(1) < m(3), where m = multiplicity; e.g., a(7) counts these 3 partitions: [4, 3], [3, 3, 1], [3, 2, 2]. - Clark Kimberling, Apr 01 2014
The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Apr 07 2014

			a(8) = 3 counts the 4's in 8 = 4+4 = 4+2+2. The 4's in 8 = 4+3+1 = 4+2+1+1 = 4+1+1+1+1 do not count.
From _Omar E. Pol_, Oct 25 2012: (Start)
--------------------------------------
Last section                   Number
of the set of                    of
partitions of 8                 4's
--------------------------------------
8 .............................. 0
4 + 4 .......................... 2
5 + 3 .......................... 0
6 + 2 .......................... 0
3 + 3 + 2 ...................... 0
4 + 2 + 2 ...................... 1
2 + 2 + 2 + 2 .................. 0
.   1 .......................... 0
.       1 ...................... 0
.       1 ...................... 0
.           1 .................. 0
.       1 ...................... 0
.           1 .................. 0
.           1 .................. 0
.               1 .............. 0
.           1 .................. 0
.               1 .............. 0
.               1 .............. 0
.                   1 .......... 0
.                   1 .......... 0
.                       1 ...... 0
.                           1 .. 0
------------------------------------
.           6 - 3 =              3
.
In the last section of the set of partitions of 8 the difference between the sum of the fourth column and the sum of the fifth column is 6 - 3 = 3 equaling the number of 4's, so a(8) = 3 (see also A024788).
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column 4 of A194812.

Cf. A015739, A024788, A135010, A138121, A182703, A182712, A182713, A240058.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<2 then [0, 0]
    else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
         [g[1]+h[1], g[2]+h[2]+`if`(i=4, h[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..70);  # Alois P. Heinz, Mar 19 2012

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 3]], {n, 0, z}] (* Clark Kimberling, Apr 01 2014 *)
b[n_, i_] := b[n, i] = Module[{g, h}, If[n==0, {1, 0}, If[i<2, {0, 0}, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[i==4, h[[1]], 0]}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 4], {n, 52}] (* Robert Price, May 15 2020 *)

A182714 = lambda n: sum(list(p).count(4) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3) + a(n+4), n >= 0. - Omar E. Pol, Feb 04 2012

A207383 Triangle read by rows: T(n,k) is the sum of parts of size k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 0, 3, 3, 4, 0, 4, 5, 2, 3, 0, 5, 7, 8, 6, 4, 0, 6, 11, 6, 6, 4, 5, 0, 7, 15, 16, 9, 12, 5, 6, 0, 8, 22, 14, 18, 8, 10, 6, 7, 0, 9, 30, 30, 18, 20, 15, 12, 7, 8, 0, 10, 42, 30, 30, 20, 20, 12, 14, 8, 9, 0, 11, 56, 54, 42, 40, 25, 30, 14, 16, 9, 10, 0, 12

Offset: 1

Omar E. Pol, Feb 24 2012

For further properties of this triangle see also A182703.

			Triangle begins:
   1;
   1,  2;
   2,  0,  3;
   3,  4,  0,  4;
   5,  2,  3,  0,  5;
   7,  8,  6,  4,  0,  6;
  11,  6,  6,  4,  5,  0,  7;
  15, 16,  9, 12,  5,  6,  0,  8;
  22, 14, 18,  8, 10,  6,  7,  0,  9;
  30, 30, 18, 20, 15, 12,  7,  8,  0, 10;
  42, 30, 30, 20, 20, 12, 14,  8,  9,  0, 11;
  56, 54, 42, 40, 25, 30, 14, 16,  9, 10,  0, 12;
...
From _Omar E. Pol_, Nov 28 2020: (Start)
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 triangle A182703
.      * * * * * * *
.      1,2,3,4,5,6,7 --> Row 7 of triangle A002260
.      = = = = = = =
.     11,6,6,4,5,0,7 --> 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 the 7th row of triangle is [11, 6, 6, 4, 5, 0, 7]. (End)

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

Column 1 is A000041.
Leading diagonal gives A000027.
Second diagonal gives A000007.
Row sums give A138879.
Cf. A002260, A066186, A135010, A138121, A138785, A138880, A182703, A194812, A207031.

T(n,k) = k*A182703(n,k).

A194702 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (2 + m).

Original entry on oeis.org

2, 0, 2, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 2. For further information see A182703 and A135010.

			Triangle begins:
2,
0, 2,
1, 0, 1,
0, 1, 0, 1,
0, 0, 1, 0, 1,
0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1, 0, 1,
...
For k = 1 and  m = 1; T(1,1) = 2 because there are two parts of size 1 in the last section of the set of partitions of 3, since 2 + m = 3, so a(1) = 2. For k = 2 and m = 1; T(2,1) = 0 because there are no parts of size 2 in the last section of the set of partitions of 3, since 2 + m = 3, so a(2) = 0.

Always the sum of row k = p(2) = A000041(n) = 2.
The first (0-10) members of this family of triangles are A023531, A129186, this sequence, A194703-A194710.
Cf. A135010, A138121, A182712-A182714, A194812.

T(k,m) = A182703(2+m,k), with T(k,m) = 0 if k > 2+m.

T(k,m) = A194812(2+m,k).

A194710 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (10 + m).

Original entry on oeis.org

42, 15, 27, 10, 14, 18, 5, 10, 10, 17, 4, 5, 8, 10, 15, 2, 5, 4, 8, 9, 14, 2, 2, 4, 5, 7, 9, 13, 1, 2, 2, 4, 4, 8, 8, 13, 1, 1, 2, 2, 4, 4, 7, 9, 12, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of row k is also the number of partitions of 10. For further information see A182703 and A135010.

			Triangle begins:
  42;
  15, 27;
  10, 14, 18;
   5, 10, 10, 17;
   4,  5,  8, 10, 15;
   2,  5,  4,  8,  9, 14;
   2,  2,  4,  5,  7,  9, 13;
   1,  2,  2,  4,  4,  8,  8, 13;
   1,  1,  2,  2,  4,  4,  7,  9, 12;
   0,  1,  1,  2,  2,  4,  4,  7,  8, 13;
   1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
   0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
   0,  0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
   0,  0,  0,  1,  0,  1,  1,  2,  2,  4,  4,  7,  8, 12;
  ...
For k = 1 and m = 1; T(1,1) = 42 because there are 42 parts of size 1 in the last section of the set of partitions of 11, since 10 + m = 11, so a(1) = 42. For k = 2 and m = 1; T(2,1) = 15 because there are 15 parts of size 2 in the last section of the set of partitions of 11, since 10 + m = 11, so a(2) = 15.

Always the sum of row k = p(10) = A000041(10) = 42.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194709, this sequence.
Cf. A002865, A135010, A138121, A194812.

T(k,m) = A182703(10+m,k), with T(k,m) = 0 if k > 10+m.
T(k,m) = A194812(10+m,k).
Beginning with row k=11 each row starts with (k-11) 0's and ends with the subsequence 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, the initial terms of A002865. - Alois P. Heinz, Feb 15 2012

A207378 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nonincreasing order.

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

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

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

Triangle similar to A138121. Mirror of A207377. Row n has length A138137(n). Row sums give A138879. Column 1 is A000027.

Cf. A006128, A066186, A135010, A182703, A182712-A182714, A194812.

A182713 Number of 3's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 2, 3, 6, 6, 10, 14, 18, 24, 35, 42, 58, 76, 97, 124, 164, 202, 261, 329, 412, 514, 649, 795, 992, 1223, 1503, 1839, 2262, 2741, 3346, 4056, 4908, 5919, 7150, 8568, 10297, 12320, 14721, 17542, 20911, 24808, 29456, 34870, 41232, 48652, 57389

Offset: 1

Omar E. Pol, Nov 28 2010

nonn

Also number of 3's in all partitions of n that do not contain 1 as a part.
Also 0 together with the first differences of A024787. - Omar E. Pol, Nov 13 2011
a(n) is the number of partitions of n having fewer 1s than 2s; e.g., a(7) counts these 3 partitions: [5, 2], [3, 2, 2], [2, 2, 2, 1]. - Clark Kimberling, Mar 31 2014
The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Apr 07 2014

			a(7) = 2 counts the 3's in 7 = 4+3 = 3+2+2. The 3's in 7 = 3+3+1 = 3+2+1+1 = 3+1+1+1+1 do not count.
From _Omar E. Pol_, Oct 27 2012: (Start)
--------------------------------------
Last section                   Number
of the set of                    of
partitions of 7                 3's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 1
5 + 2 .......................... 0
3 + 2 + 2 ...................... 1
.   1 .......................... 0
.       1 ...................... 0
.       1 ...................... 0
.           1 .................. 0
.       1 ...................... 0
.           1 .................. 0
.           1 .................. 0
.               1 .............. 0
.               1 .............. 0
.                   1 .......... 0
.                       1 ...... 0
------------------------------------
.       5 - 3 =                  2
.
In the last section of the set of partitions of 7 the difference between the sum of the third column and the sum of the fourth column is 5 - 3 = 2 equaling the number of 3's, so a(7) = 2 (see also A024787).
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column 3 of A194812.

Cf. A135010, A138121, A174455, A182703, A182712, A182714, A240056.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<2 then [0, 0]
    else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
         [g[1]+h[1], g[2]+h[2]+`if`(i=3, h[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 18 2012

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 2]], {n, 0, z}] (* Clark Kimberling, Mar 31 2014 *)
b[n_, i_] := b[n, i] = Module[{g, h}, If[n == 0, {1, 0}, If[i<2, {0, 0}, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; Join[g[[1]] + h[[1]], g[[2]] + h[[2]] + If[i == 3, h[[1]], 0]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Nov 30 2015, after Alois P. Heinz *)
Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 3], {n, 51}] (* Robert Price, May 15 2020 *)

A182713 = lambda n: sum(list(p).count(3) for p in Partitions(n) if 1 not in p) # D. S. McNeil, Nov 29 2010

It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3), n >= 0. - Omar E. Pol, Feb 04 2012

a(n) ~ A000041(n)/3 ~ exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n). - Vaclav Kotesovec, Jan 03 2019

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

cp's OEIS Frontend

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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A182712 Number of 2's in the last section of the set of partitions of n.

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A182714 Number of 4's in the last section of the set of partitions of n.

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A207383 Triangle read by rows: T(n,k) is the sum of parts of size k in the last section of the set of partitions of n.

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A194702 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (2 + m).

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A194710 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (10 + m).

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A207378 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nonincreasing order.

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