A182712 -id:A182712 - cp's OEIS frontend

A182722 a(n) = A005291(n+1)-A182712(n).

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 14, 26, 36, 60, 83, 128, 175, 261, 351, 504, 674, 943, 1247, 1711, 2243

Offset: 0

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

nonn

The difference between two apparently unrelated sequences which happen to have the same initial terms. - N. J. A. Sloane, Dec 01 2010

Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182712.

a(n) = A005291(n+1)-A182712(n)

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

A194812 Square array read by antidiagonals: T(n,k) = number of parts of size k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 0, 0, 0, 5, 2, 1, 0, 0, 7, 1, 0, 0, 0, 0, 11, 4, 1, 1, 0, 0, 0, 15, 3, 2, 0, 0, 0, 0, 0, 22, 8, 2, 1, 1, 0, 0, 0, 0, 30, 7, 3, 1, 0, 0, 0, 0, 0, 0, 42, 15, 6, 3, 1, 1, 0, 0, 0, 0, 0, 56, 15, 6, 2, 1, 0, 0, 0, 0, 0, 0, 0, 77, 27, 10

Offset: 1

Omar E. Pol, Feb 04 2012

It appears that in the column k, starting in row n, the sum of k successive terms is equal to A000041(n-1).

			Array begins:
.  1,  0,  0,  0, 0, 0, 0, 0, 0, 0, 0, 0,...
.  1,  1,  0,  0, 0, 0, 0, 0, 0, 0, 0, 0,...
.  2,  0,  1,  0, 0, 0, 0, 0, 0, 0, 0, 0,...
.  3,  2,  0,  1, 0, 0, 0, 0, 0, 0, 0, 0,...
.  5,  1,  1,  0, 1, 0, 0, 0, 0, 0, 0, 0,...
.  7,  4,  2,  1, 0, 1, 0, 0, 0, 0, 0, 0,...
. 11,  3,  2,  1, 1, 0, 1, 0, 0, 0, 0, 0,...
. 15,  8,  3,  3, 1, 1, 0, 1, 0, 0, 0, 0,...
. 22,  7,  6,  2, 2, 1, 1, 0, 1, 0, 0, 0,...
. 30, 15,  6,  5, 3, 2, 1, 1, 0, 1, 0, 0,...
. 42, 15, 10,  5, 4, 2, 2, 1, 1, 0, 1, 0,...
. 56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1,...
...
For n = 7, from the conjecture we have that p(n-1) = p(6) = 11 = 3+8 = 2+3+6 = 1+3+2+5 = 1+1+2+3+4 = 0+1+1+2+2+5, etc. where p(n) = A000041(n).

Columns 1-4: A000041, A182712, A182713, A182714. Main triangle: A182703.

Cf. A066633, A135010, A138121, A138137.

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

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

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).

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.

A100818 For a given unrestricted partition pi, let P(pi)=lambda(pi), if mu(pi)=0. If mu(pi)>0 then let P(pi)=nu(pi), where nu(pi) is the number of parts of pi greater than mu(pi), mu(pi) is the number of ones in pi and lambda(pi) is the largest part of pi.

Original entry on oeis.org

1, 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, 79864

Offset: 1

David S. Newman, Jan 13 2005

nonn

Note that this is very similar to the "crank" of Andrews and Garvan. The number of partitions pi with P(pi) odd is the given sequence.
The sequence is the same as A087787 except for the value of a(1) (this was established by George Andrews, Jan 18 2005). If "even" is replace by "odd" in the definition of the sequence, the new sequence is almost identical except for two values and a shift to the right.
Also, positive integers of A182712. a(n) is also the number of 2's in the n-th row that contain a 2 as a part in the triangle of A138121 (note that rows 1 and 3 do not contain a 2 as a part). - Omar E. Pol, Nov 28 2010

			a(3)=1 because P(3)=3, P(2 1)=1 and P(1 1 1)=0.

G. C. Greubel, Table of n, a(n) for n = 1..1000
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.

Cf. A000041, A087787, A135010, A138121, A182712.

Rest[ CoefficientList[ Series[x + 1/(1 + x) Product[1/(1 - x^n), {n, 50}], {x, 0, 50}], x]] (* Robert G. Wilson v, Feb 11 2005 *)

G.f.: x+(1/(1+x))* Product_{n>=1}(1/(1-x^n)). [corrected by Vaclav Kotesovec, Aug 29 2019]
a(n) = A000041(n) - a(n-1), for n>2. - Jon Maiga, Aug 29 2019 [corrected by Vaclav Kotesovec, Aug 29 2019]
a(n) = a(n-2) + A000041(n-1) - A000041(n-2), for n>=3. - Vaclav Kotesovec, Aug 29 2019

More terms from Robert G. Wilson v, Feb 11 2005

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

A005291 Representation degeneracies for boson strings.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 4, 3, 8, 7, 15, 15, 28, 30, 51, 58, 92, 108, 163, 196, 285, 348, 490, 605, 833, 1034, 1396, 1740, 2313, 2887, 3789, 4730, 6141, 7661, 9853, 12276, 15664, 19477, 24678, 30621, 38558, 47728, 59767, 73795, 91949, 113239, 140452

Offset: 1

N. J. A. Sloane

nonn

This gives the number of vector ([1]) irreducible representations in the n-th excited state of an oscillator model of bosons. A005290 counts the scalar ([0]) representations, A005292 the rank-2 tensor representations ([2]). - R. J. Mathar, Nov 30 2010

a(2)..a(12) coincide with A182712(n-1). - Omar E. Pol, Feb 07 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. L. Curtright and C. B. Thorn, Symmetry patterns in the mass spectra of dual string models, Nuclear Phys. B 274 (1986), 520-588.

G3(1); # Using the code in A005290. - Sean A. Irvine, Apr 30 2016

More terms from Sean A. Irvine, Apr 30 2016

cp's OEIS Frontend

A182722 a(n) = A005291(n+1)-A182712(n).

Views

Author

Keywords

Comments

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A194812 Square array read by antidiagonals: T(n,k) = number of parts of size k in the last section of the set of partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A100818 For a given unrestricted partition pi, let P(pi)=lambda(pi), if mu(pi)=0. If mu(pi)>0 then let P(pi)=nu(pi), where nu(pi) is the number of parts of pi greater than mu(pi), mu(pi) is the number of ones in pi and lambda(pi) is the largest part of pi.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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