A207032 -id:A207032 - cp's OEIS frontend

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

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

A138137 First differences of A006128.

Original entry on oeis.org

1, 2, 3, 6, 8, 15, 19, 32, 42, 64, 83, 124, 157, 224, 288, 395, 502, 679, 854, 1132, 1422, 1847, 2307, 2968, 3677, 4671, 5772, 7251, 8908, 11110, 13572, 16792, 20439, 25096, 30414, 37138, 44798, 54389, 65386, 78959, 94558, 113687, 135646, 162375, 193133

Offset: 1

Omar E. Pol, Mar 18 2008

Number of parts in the last section of the set of partitions of n (see A135010, A138121).
Sum of largest parts in all partitions in the head of the last section of the set of partitions of n. - Omar E. Pol, Nov 07 2011
From Omar E. Pol, Feb 16 2021: (Start)
Convolution of A341062 and A000041.
Convolution of A000005 and A002865.
a(n) is also the total number of parts in the n-th section of the set of partitions of any positive integer >= n.
a(n) is also the total number of divisors of all terms in the n-th row of triangle A336811. These divisors are also all parts in the last section of the set of partitions of n. (End)

			From _Omar E. Pol_, Feb 19 2012: (Start)
Illustration of initial terms (n = 1..6) as sums of the first columns from the last sections of the first six natural numbers (or from the first six sections of 6):
.                                           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,  2,    3,      6,        8,          15,
...
Also, we can see that the sequence gives the number of parts in each section. For the number of odd/even parts (and more) see A207031, A207032 and also A206563. (End)
From _Omar E. Pol_, Aug 16 2013: (Start)
The geometric model looks like this:
.                                           _ _ _ _ _ _
.                                          |_ _ _ _ _ _|
.                                          |_ _ _|_ _ _|
.                                          |_ _ _ _|_ _|
.                               _ _ _ _ _  |_ _|_ _|_ _|
.                              |_ _ _ _ _|           |_|
.                     _ _ _ _  |_ _ _|_ _|           |_|
.                    |_ _ _ _|         |_|           |_|
.             _ _ _  |_ _|_ _|         |_|           |_|
.       _ _  |_ _ _|       |_|         |_|           |_|
.   _  |_ _|     |_|       |_|         |_|           |_|
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1    2      3        6          8           15
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the divisors of these terms are [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]. There are 15 divisors so a(6) = 15. - _Omar E. Pol_, Jul 27 2021

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

Column 1 of A207031.

Cf. A000005, A000041, A002865, A006128, A135010, A138121, A182703, A336811, A336812, A338156, A340035, A341062.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 19 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]]+g[[1]], f[[2]]+g[[2]]+g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
Table[PartitionsP[n - 1] + Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 45}] (* Robert Price, May 01 2020 *)

a(n) = A006128(n) - A006128(n-1).
a(n) = A000041(n-1) + A138135(n). - Omar E. Pol, Nov 07 2011
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(6*n/Pi^2)) / (8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 21 2016
G.f.: Sum_{i>=1} i*x^i * Product_{j=2..i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 04 2017

A207031 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the last section of the set of partitions of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 6, 3, 1, 1, 8, 3, 2, 1, 1, 15, 8, 4, 2, 1, 1, 19, 8, 5, 3, 2, 1, 1, 32, 17, 9, 6, 3, 2, 1, 1, 42, 20, 13, 7, 5, 3, 2, 1, 1, 64, 34, 19, 13, 8, 5, 3, 2, 1, 1, 83, 41, 26, 16, 11, 7, 5, 3, 2, 1, 1, 124, 68, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Feb 14 2012

Also T(n,k) is the number of parts >= k in the last section of the set of partitions of n. Therefore T(n,1) = A138137(n), the total number of parts in the last section of the set of partitions of n. For calculation of the number of odd/even parts, etc, follow the same rules from A206563.
More generally, let m and n be two positive integers such that m <= n. It appears that any set formed by m connected sections, or m disconnected sections, or a mixture of both, has the same properties described in the entry A206563.
It appears that reversed rows converge to A000041.
It appears that the first differences of row n together with 1 give the row n of triangle A182703 (see example). - Omar E. Pol, Feb 26 2012

			Illustration of initial terms. First six rows of triangle as sums of columns from the last sections of the first six natural numbers (or as sums of columns from the six sections of 6):
.                                         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
. --- --- ------- --------- ----------- --------------
A: 1, 2,1, 3,1,1,  6,3,1,1,  8,3,2,1,1,  15,8,4,2,1,1
.  |  |/|  |/|/|   |/|/|/|   |/|/|/|/|    |/|/|/|/|/|
B: 1, 1,1, 2,0,1,  3,2,0,1,  5,1,1,0,1,   7,4,2,1,0,1
.
A := initial terms of this triangle.
B := initial terms of triangle A182703.
.
Triangle begins:
1;
2,    1;
3,    1,  1;
6,    3,  1,  1;
8,    3,  2,  1,  1;
15,   8,  4,  2,  1,  1;
19,   8,  5,  3,  2,  1,  1;
32,  17,  9,  6,  3,  2,  1,  1;
42,  20, 13,  7,  5,  3,  2,  1,  1;
64,  34, 19, 13,  8,  5,  3,  2,  1,  1;
83,  41, 26, 16, 11,  7,  5,  3,  2,  1,  1;
124, 68, 41, 27, 17, 12,  7,  5,  3,  2,  1,  1;

Columns (1-2): A138137, A138135.
Row sums give A138879.
Cf. A000041, A002865, A006128, A135010, A138121, A181187, A182703, A206562, A206563, A207032, A207379, A208476, A210955, A210956.

From Omar E. Pol, Dec 07 2019: (Start)
From the formula in A138135 (year 2008) we have that:
A000041(n-1) = A138137(n) - A138135(n) = T(n,1) - T(n,2);
Hence A000041(n) = T(n+1,1) - T(n+1,2), n >= 0;
Also  A000041(n) = A002865(n) + T(n,1) - T(n,2). (End)

More terms from Alois P. Heinz, Feb 17 2012

A206563 Triangle read by rows: T(n,k) = number of odd/even parts >= k in all partitions of n, if k is odd/even.

Original entry on oeis.org

1, 2, 1, 5, 1, 1, 8, 4, 1, 1, 15, 5, 3, 1, 1, 24, 11, 5, 3, 1, 1, 39, 15, 9, 4, 3, 1, 1, 58, 28, 13, 9, 4, 3, 1, 1, 90, 38, 23, 12, 8, 4, 3, 1, 1, 130, 62, 33, 21, 12, 8, 4, 3, 1, 1, 190, 85, 51, 29, 20, 11, 8, 4, 3, 1, 1, 268, 131, 73, 48, 28, 20, 11, 8, 4, 3, 1, 1

Offset: 1

Omar E. Pol, Feb 15 2012

Let m and n be two positive integers such that m <= n. It appears that any set formed by m connected sections, or m disconnected sections, or a mixture of both, has the same properties described in the section example. (Cf. A135010, A207031, A207032, A212010). - Omar E. Pol, May 01 2012

			Calculation for n = 6. Write the partitions of 6 and below the sums of their columns:
.
.   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
. ------------------------
.  35, 16,  8,  4,  2,  1  --> Row 6 of triangle A181187.
.   |  /|  /|  /|  /|  /|
.   | / | / | / | / | / |
.   |/  |/  |/  |/  |/  |
.  19,  8,  4,  2,  1,  1  --> Row 6 of triangle A066633.
.
More generally, it appears that the sum of column k is also the total number of parts >= k in all partitions of n. It appears that the first differences of the column sums together with 1 give the number of occurrences of k in all partitions of n.
On the other hand we can see that the partitions of 6 contain:
24  odd parts >= 1 (the odd parts).
11 even parts >= 2 (the even parts).
5   odd parts >= 3.
3  even parts >= 4.
2   odd parts >= 5.
1  even part  >= 6.
Then, using the values of the column sums, it appears that:
T(6,1) = 35 - 16 + 8 - 4 + 2 - 1 = 24
T(6,2) =      16 - 8 + 4 - 2 + 1 = 11
T(6,3) =           8 - 4 + 2 - 1 = 5
T(6,4) =               4 - 2 + 1 = 3
T(6,5) =                   2 - 1 = 1
T(6,6) =                       1 = 1
So the 6th row of our triangle gives 24, 11, 5, 3, 1, 1.
Finally, for all partitions of 6, we can write:
The number of  odd parts      is equal to T(6,1) = 24.
The number of even parts      is equal to T(6,2) = 11.
The number of  odd parts >= 3 is equal to T(6,3) = 5.
The number of even parts >= 4 is equal to T(6,4) = 3.
The number of  odd parts >= 5 is equal to T(6,5) = 1.
The number of even parts >= 6 is equal to T(6,6) = 1.
More generally, we can write the same properties for any positive integer.
Triangle begins:
1;
2,    1;
5,    1,  1;
8,    4,  1,  1;
15,   5,  3,  1,  1;
24,  11,  5,  3,  1,  1;
39,  15,  9,  4,  3,  1,  1;
58,  28, 13,  9,  4,  3,  1,  1;
90,  38, 23, 12,  8,  4,  3,  1,  1;
130, 62, 33, 21, 12,  8,  4,  3,  1,  1;

Columns 1-2 give A066897, A066898.

Cf. A006128, A066633, A181187, A182703, A207031, A207032.

It appears that T(n,k) = abs(Sum_{j=k..n} (-1)^j*A181187(n,j)).

It appears that A066633(n,k) = T(n,k) - T(n,k+2). - Omar E. Pol, Feb 26 2012

More terms from Alois P. Heinz, Feb 18 2012

A207034 Sum of all parts minus the number of parts of the n-th partition in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8, 5, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 9, 6, 7, 7, 8, 7, 8, 8, 9, 8, 8, 9, 9, 9, 10, 6, 7, 7, 8, 8, 8, 9, 8, 9, 9, 9, 10, 8, 9, 9, 10, 9, 10, 10, 10, 11, 7, 8, 8, 9, 8, 9

Offset: 1

Omar E. Pol, Feb 20 2012

a(n) is also the column number in which is located the part of size 1 in the n-th zone of the tail of the last section of the set of partitions of k in colexicographic order, minus the column number in which is located the part of size 1 in the first row of the same tail, when k -> infinity (see example). For the definition of "section" see A135010.

			Illustration of initial terms, n = 1..15. Consider the last 15 rows of the tail of the last section of the set of partitions in colexicographic order of any integer >= 8. The tail contains at least A000041(8-1) = 15 parts of size 1. a(n) is also the number of dots in the n-th row of the diagram.
----------------------------------
n      Tail                  a(n)
----------------------------------
15        1 . . . . . .       6
14          1 . . . . .       5
13          1 . . . . .       5
12            1 . . . .       4
11          1 . . . . .       5
10            1 . . . .       4
9             1 . . . .       4
8               1 . . .       3
7             1 . . . .       4
6               1 . . .       3
5               1 . . .       3
4                 1 . .       2
3                 1 . .       2
2                   1 .       1
1                     1       0
----------------------------------
Written as a triangle:
0;
1;
2;
2,3;
3,4;
3,4,4,5;
4,5,5,6;
4,5,5,6,6,6,7;
5,6,6,7,6,7,7,8;
5,6,6,7,7,7,8,7,8,8,8,9;
6,7,7,8,7,8,8,9,8,8,9,9,9,10;
6,7,7,8,8,8,9,8,9,9,9,10,8,9,9,10,9,10,10,10,11;
...
Consider a matrix [j X A000041(j)] in which the rows represent the partitions of j in colexicographic order (see A211992). Every part of every partition is located in a cell of the matrix. We can see that a(n) is the number of empty cells in row n for any integer j, if A000041(j) >= n. The number of empty cells in row n equals the sum of all parts minus the number of parts in the n-th partition of j.
Illustration of initial terms. The smallest part of every partition is located in the last column of the matrix.
---------------------------------------------------------
.   j: 1    2       3         4           5             6
n a(n)
---------------------------------------------------------
1  0 | 1  1 1   1 1 1   1 1 1 1   1 1 1 1 1   1 1 1 1 1 1
2  1 |    . 2   . 2 1   . 2 1 1   . 2 1 1 1   . 2 1 1 1 1
3  2 |          . . 3   . . 3 1   . . 3 1 1   . . 3 1 1 1
4  2 |                  . . 2 2   . . 2 2 1   . . 2 2 1 1
5  3 |                  . . . 4   . . . 4 1   . . . 4 1 1
6  3 |                            . . . 3 2   . . . 3 2 1
7  4 |                            . . . . 5   . . . . 5 1
8  3 |                                        . . . 2 2 2
9  4 |                                        . . . . 4 2
10 4 |                                        . . . . 3 3
11 5 |                                        . . . . . 6
...
Illustration of initial terms. In this case the largest part of every partition is located in the first column of the matrix.
---------------------------------------------------------
.   j: 1    2       3         4           5             6
n a(n)
---------------------------------------------------------
1  0 | 1  1 1   1 1 1   1 1 1 1   1 1 1 1 1   1 1 1 1 1 1
2  1 |    2 .   2 1 .   2 1 1 .   2 1 1 1 .   2 1 1 1 1 .
3  2 |          3 . .   3 1 . .   3 1 1 . .   3 1 1 1 . .
4  2 |                  2 2 . .   2 2 1 . .   2 2 1 1 . .
5  3 |                  4 . . .   4 1 . . .   4 1 1 . . .
6  3 |                            3 2 . . .   3 2 1 . . .
7  4 |                            5 . . . .   5 1 . . . .
8  3 |                                        2 2 2 . . .
9  4 |                                        4 2 . . . .
10 4 |                                        3 3 . . . .
11 5 |                                        6 . . . . .
...

Alois P. Heinz, Table of n, a(n) for n = 1..10143

Row r has length A187219(r). Partial sums give A207038. Row sums give A207035. Right border gives A001477. Where records occur give A000041 without repetitions.

Cf. A135010, A138121, A141285, A182703, A194548, A196087, A207031, A207032, A207035, A211992, A228716, A230440.

a(n) = t(n) - A194548(n), if n >= 2, where t(n) is the n-th element of the following sequence: triangle read by rows in which row n lists n repeated k times, where k = A187219(n).
a(n) = A000120(A194602(n-1)) = A000120(A228354(n)-1).
a(n) = i - A193173(i,n), i >= 1, 1<=n<=A000041(i).

A206562 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, 3, 2, 5, 3, 3, 11, 8, 4, 4, 15, 10, 8, 5, 5, 31, 24, 16, 10, 6, 6, 39, 28, 22, 16, 12, 7, 7, 71, 56, 40, 31, 19, 14, 8, 8, 94, 72, 58, 40, 32, 22, 16, 9, 9, 150, 120, 90, 72, 52, 37, 25, 18, 10, 10, 196, 154, 124, 94, 74, 54, 42, 28, 20, 11, 11

Offset: 1

Omar E. Pol, Feb 15 2012

			Triangle begins:
1;
3,   2;
5,   3,  3;
11,  8,  4,  4;
15, 10,  8,  5,  5;
31, 24, 16, 10,  6,  6;
39, 28, 22, 16, 12,  7,  7;
71, 56, 40, 31, 19, 14,  8,  8;
94, 72, 58, 40, 32, 22, 16,  9,  9;

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

Columns 1-2 give A138879, A138880. Diagonal is A000027.

Cf. A135010, A138121, A182703, A206561, A207031, A207032.

A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 6, 9, 11, 12, 8, 14, 17, 19, 20, 15, 23, 29, 32, 34, 35, 19, 34, 42, 48, 51, 53, 54, 32, 51, 66, 74, 80, 83, 85, 86, 42, 74, 93, 108, 116, 122, 125, 127, 128, 64, 106, 138, 157, 172, 180, 186, 189, 191, 192, 83, 147, 189, 221, 240

Offset: 1

Omar E. Pol, Apr 26 2012

The set of partitions of n contains n shells (see A135010). Let m and n be two positive integers such that m <= n. It appears that in any set formed by m connected shells, or m disconnected shells, or a mixture of both, the sum of all parts of the j-th column equals the total number of parts >= j in the same set (see example). More generally it appears that any of these sets has the same properties mentioned in A206563 and A207031.

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 the first column equals the total number of parts 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
. ---------- ---------- ---------- ---------- ----------
.  8         14         17         19         20
.
So row 5 lists 8, 14, 17, 19, 20.
.
Triangle begins:
1;
2,    3;
3,    5,   6;
6,    9,  11,  12;
8,   14,  17,  19,  20;
15,  23,  29,  32,  34,  35;
19,  34,  42,  48,  51,  53,  54;
32,  51,  66,  74,  80,  83,  85,  86;
42,  74,  93, 108, 116, 122, 125, 127, 128;
64, 106, 138, 157, 172, 180, 186, 189, 191, 192;

Mirror of triangle A212000. Column 1 is A138137. Right border is A006128.

Cf. A135010, A138121, A181187, A182703, A206563, A207031, A207032, A212001, A212011

T(n,k) = A006128(n) - A006128(n-k).

T(n,k) = Sum_{j=n-k+1..n} A138137(j).

A207381 Total sum of the odd-indexed parts of all partitions of n.

Original entry on oeis.org

1, 3, 7, 14, 25, 45, 72, 117, 180, 275, 403, 596, 846, 1206, 1681, 2335, 3183, 4342, 5820, 7799, 10321, 13622, 17798, 23221, 30009, 38706, 49567, 63316, 80366, 101805, 128211, 161134, 201537, 251495, 312508, 387535, 478674, 590072, 724920, 888795, 1086324

Offset: 1

Omar E. Pol, Feb 17 2012

nonn

For more information see A206563.

			For n = 5, write the partitions of 5 and below write the sums of their odd-indexed parts:
.    5
.    3+2
.    4+1
.    2+2+1
.    3+1+1
.    2+1+1+1
.    1+1+1+1+1
.  ------------
.   20 + 4 + 1 = 25
The total sum of the odd-indexed parts is 25 so a(5) = 25.

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

Cf. A066186, A066897, A066898, A181187, A194714, A206283, A206563, A207031, A207032, A207382.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0$2]
    elif i<1 then [0$3]
    else g:= b(n, i-1); h:= `if`(i>n, [0$3], b(n-i, i));
         [g[1]+h[1], g[2]+h[3], g[3]+h[2]+i*h[1]]
      fi
    end:
a:= n-> b(n,n)[3]:
seq(a(n), n=1..50); # Alois P. Heinz, Mar 12 2012

b[n_, i_] := b[n, i] = Module[{g, h}, If[n == 0 , {1, 0, 0}, If[i < 1, {0, 0, 0},  g = b[n, i - 1]; h = If[i > n, {0, 0, 0}, b[n - i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[3]], g[[3]] + h[[2]] + i*h[[1]]}]]]; a[n_] := b[n, n][[3]]; Table [a[n], {n, 1, 50}] (* Jean-François Alcover, Dec 09 2016 after Alois P. Heinz *)

a(n) = A066186(n) - A207382(n) = A066897(n) + A207382(n).

More terms from Alois P. Heinz, Mar 12 2012

A207382 Sum of the even-indexed parts of all partitions of n.

Original entry on oeis.org

0, 1, 2, 6, 10, 21, 33, 59, 90, 145, 213, 328, 467, 684, 959, 1361, 1866, 2588, 3490, 4741, 6311, 8422, 11067, 14579, 18941, 24630, 31703, 40788, 52019, 66315, 83891, 106034, 133182, 167045, 208397, 259637, 321895, 398498, 491295, 604725, 741579, 908008

Offset: 1

Omar E. Pol, Feb 17 2012

nonn

Also the sum of the floors of half the parts of all partitions of n, because the sum of one kind for a partition equals the sum of the other kind for the conjugate partition. Furthermore, this generalizes to taking m-th indices and dividing by m. - George Beck, Apr 15 2017

			For n = 5, write the partitions of 5 and below write the sums of their even-indexed parts:
. 5
. 3+2
. 4+1
. 2+2+1
. 3+1+1
. 2+1+1+1
. 1+1+1+1+1
------------
.   8 + 2   = 10
The sum of the even-indexed parts is 10, so a(5) = 10.
From _George Beck_, Apr 15 2017: (Start)
Alternatively, sum the floors of the parts divided by 2:
. 2
. 1+1
. 2+0
. 1+1+0
. 1+0+0
. 1+0+0+0
. 0+0+0+0+0
The sum is 10, so a(5) = 10. (End)

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

For more information see A206563.

Cf. A066186, A066897, A066898, A181187, A194714, A206283, A207031, A207032, A207381.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0$2]
    elif i<1 then [0$3]
    else g:= b(n, i-1); h:= `if`(i>n, [0$3], b(n-i, i));
         [g[1]+h[1], g[2]+h[3], g[3]+h[2]+i*h[1]]
      fi
    end:
a:= n-> b(n,n)[2]:
seq (a(n), n=1..50); # Alois P. Heinz, Mar 12 2012

b[n_, i_] := b[n, i] = Module[{g, h}, Which[n==0, {1, 0, 0}, i<1, {0, 0, 0}, True, g = b[n, i-1]; h = If[i>n, {0, 0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[3]], g[[3]] + h[[2]] + i*h[[1]]}]]; a[n_] := b[n, n][[2]]; Table [a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
a[n_]:= Total@Flatten@Quotient[IntegerPartitions[n], 2];
Table [a[n], {n, 1, 50}] (* George Beck, Apr 15 2017 *)

a(n) = A066186(n) - A207381(n) = A207381(n) - A066897(n).

More terms from Alois P. Heinz, Mar 12 2012

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

cp's OEIS Frontend

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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A138137 First differences of A006128.

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A207031 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the last section of the set of partitions of n.

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A206563 Triangle read by rows: T(n,k) = number of odd/even parts >= k in all partitions of n, if k is odd/even.

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A207034 Sum of all parts minus the number of parts of the n-th partition in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j).

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A206562 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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A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n.

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A207381 Total sum of the odd-indexed parts of all partitions of n.

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