A194714 -id:A194714 - cp's OEIS frontend

A206283 Triangle read by rows: T(n,k) = sum of the k-th parts of all partitions of n with their parts written in nondecreasing order.

1, 3, 1, 5, 3, 1, 9, 7, 3, 1, 12, 12, 7, 3, 1, 20, 21, 14, 7, 3, 1, 25, 31, 24, 14, 7, 3, 1, 38, 47, 40, 26, 14, 7, 3, 1, 49, 66, 61, 43, 26, 14, 7, 3, 1, 69, 93, 92, 70, 45, 26, 14, 7, 3, 1, 87, 124, 130, 106, 73, 45, 26, 14, 7, 3, 1

Offset: 1

Omar E. Pol, Feb 13 2012

In row n, the sum of all odd-indexed terms minus the sum of all even-indexed terms is equal to A194714(n).

Reversed rows converge to A014153. - Alois P. Heinz, Feb 13 2012

			Row 4 is 9, 7, 3, 1 because the five partitions of 4, with their parts written in nondecreasing order, are
.                               4
.                               1, 3
.                               2, 2
.                               1, 1, 2
.                               1, 1, 1, 1
-------------------------------------------
And the sums of the columns are 9, 7, 3, 1.
.
Triangle begins:
   1;
   3,  1;
   5,  3,  1;
   9,  7,  3,  1;
  12, 12,  7,  3,  1;
  20, 21, 14,  7,  3,  1;
  25, 31, 24, 14,  7,  3,  1;
  38, 47, 40, 26, 14,  7,  3,  1;
  49, 66, 61, 43, 26, 14,  7,  3,  1;
  69, 93, 92, 70, 45, 26, 14,  7,  3,  1;

Alois P. Heinz, Rows n = 1..70, flattened

Column 1 is A046746. Row sums give A066186.

Cf. A014153, A181187, A194714.

More terms from Alois P. Heinz, Feb 13 2012

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

A210950 Triangle read by rows: T(n,k) = number of parts in the k-th column of the partitions of n but with the partitions aligned to the right margin.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 4, 6, 7, 1, 2, 4, 7, 10, 11, 1, 2, 4, 7, 11, 14, 15, 1, 2, 4, 7, 12, 17, 21, 22, 1, 2, 4, 7, 12, 18, 25, 29, 30, 1, 2, 4, 7, 12, 19, 28, 36, 41, 42, 1, 2, 4, 7, 12, 19, 29, 40, 50, 55, 56, 1, 2, 4, 7, 12, 19, 30, 43

Offset: 1

Omar E. Pol, Apr 22 2012

Index of the first partition of n that has k parts, when the partitions of n are listed in reverse lexicographic order, as in Mathematica's IntegerPartitions[n]. - Clark Kimberling, Oct 16 2023

			For n = 6 the partitions of 6 aligned to the right margin look like this:
.
.                      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
.
The number of parts in columns 1-6 are
.  1,  2,  4,  7, 10, 11, the same as the 6th row of triangle.
Triangle begins:
  1;
  1, 2;
  1, 2, 3;
  1, 2, 4, 5;
  1, 2, 4, 6, 7;
  1, 2, 4, 7, 10, 11;
  1, 2, 4, 7, 11, 14, 15;
  1, 2, 4, 7, 12, 17, 21, 22;
  1, 2, 4, 7, 12, 18, 25, 29, 30;
  1, 2, 4, 7, 12, 19, 28, 36, 41, 42;
  1, 2, 4, 7, 12, 19, 29, 40, 50, 55, 56;
  1, 2, 4, 7, 12, 19, 30, 43, 58, 70, 76, 77;

Mirror of A058399. Row sums give A006128. Right border gives A000041, n >= 1. Rows converge to A000070.

Cf. A135010, A194714, A210945, A210951, A210952, A210953, A210970.

m[n_, k_] := Length[IntegerPartitions[n][[k]]]; c[n_] := PartitionsP[n];
t[n_, h_] := Select[Range[c[n]], m[n, #] == h &, 1];
Column[Table[t[n, h], {n, 1, 20}, {h, 1, n}]]
 (* Clark Kimberling, Oct 16 2023 *)

T(n,k) = Sum_{j=1..n} A210951(j,k).

A210951 Triangle read by rows: T(n,k) = number of parts in the k-th column of the shell model of partitions considering only the n-th shell and with its parts aligned to the right margin.

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 1, 5, 0, 0, 0, 1, 7, 0, 0, 0, 1, 3, 11, 0, 0, 0, 0, 1, 3, 15, 0, 0, 0, 0, 1, 3, 6, 22, 0, 0, 0, 0, 0, 1, 4, 7, 30, 0, 0, 0, 0, 0, 1, 3, 7, 11, 42, 0, 0, 0, 0, 0, 0, 1, 4, 9, 13, 56, 0, 0, 0, 0, 0, 0, 1, 3, 8, 15, 20, 77, 0, 0, 0

Offset: 1

Omar E. Pol, Apr 22 2012

			For n = 6 and k = 1..6 the 6th shell looks like this:
-------------------------
k: 1,  2,  3,  4,  5,  6
-------------------------
.                      6
.                  3 + 3
.                  4 + 2
.              2 + 2 + 2
.                      1
.                      1
.                      1
.                      1
.                      1
.                      1
.                      1
.
The total number of parts in columns 1-6 are
.  0,  0,  0,  1,  3, 11, the same as the 6th row of triangle.
Triangle begins:
1;
0, 2;
0, 0, 3;
0, 0, 1, 5;
0, 0, 0, 1, 7;
0, 0, 0, 1, 3, 11;
0, 0, 0, 0, 1, 3, 15;
0, 0, 0, 0, 1, 3, 6, 22;
0, 0, 0, 0, 0, 1, 4, 7, 30;
0, 0, 0, 0, 0, 1, 3, 7, 11, 42;
0, 0, 0, 0, 0, 0, 1, 4, 9, 13, 56;
0, 0, 0, 0, 0, 0, 1, 3, 8, 15, 20, 77;

Row sums give A138137. Column sums converge to A000070. Right border gives A000041, n >= 1.

Cf. A135010, A138121, A194714, A210945, A210950, A210952, A210953.

A210946 Triangle read by rows: T(n,k) = sum of parts in the k-th column of the mirror of the last section of the set of partitions of n with its parts aligned to the right margin.

Original entry on oeis.org

1, 3, 5, 9, 2, 12, 3, 20, 9, 2, 25, 11, 3, 38, 22, 9, 2, 49, 28, 14, 3, 69, 44, 26, 9, 2, 87, 55, 37, 14, 3, 123, 83, 62, 29, 9, 2, 152

Offset: 1

Omar E. Pol, Apr 21 2012

Row n lists the positive terms of the n-th row of triangle A210953 in decreasing order.

			For n = 7 the illustration shows two arrangements of the last section of the set of partitions of 7:
.
.       (7)        (7)
.     (4+3)        (3+4)
.     (5+2)        (2+5)
.   (3+2+2)        (2+2+3)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.                 ---------
.                  25,11,3
.
The left hand picture shows the last section of 7 with its parts aligned to the right margin. In the right hand picture (the mirror) we can see that the sum of all parts of the columns 1..3 are 25, 11, 3 therefore row 7 lists 25, 11, 3.
Written as a triangle begins:
1;
3;
5;
9,    2;
12,   3;
20,   9,  2;
25,  11,  3;
38,  22,  9,  2;
49,  28, 14,  3;
69,  44, 26,  9,  2;
87,  55, 37, 14,  3,
123, 83, 62, 29,  9,  2;

Column 1 is A046746, n >= 1. Row sums give A138879.

Cf. A135010, A138121, A182703, A194714, A196807, A206437, A207031, A207034, A207035, A210945, A210952, A210953.

cp's OEIS Frontend

A206283 Triangle read by rows: T(n,k) = sum of the k-th parts of all partitions of n with their parts written in nondecreasing order.

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

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Maple

Mathematica

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

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Maple

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A210950 Triangle read by rows: T(n,k) = number of parts in the k-th column of the partitions of n but with the partitions aligned to the right margin.

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A210951 Triangle read by rows: T(n,k) = number of parts in the k-th column of the shell model of partitions considering only the n-th shell and with its parts aligned to the right margin.

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A210946 Triangle read by rows: T(n,k) = sum of parts in the k-th column of the mirror of the last section of the set of partitions of n with its parts aligned to the right margin.

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