A181187 -id:A181187 - cp's OEIS frontend

A208475 Triangle read by rows: T(n,k) = total sum of odd/even parts >= k in all partitions of n, if k is odd/even.

1, 2, 2, 7, 2, 3, 10, 10, 3, 4, 23, 12, 11, 4, 5, 36, 30, 17, 14, 5, 6, 65, 40, 35, 18, 17, 6, 7, 94, 82, 49, 44, 22, 20, 7, 8, 160, 110, 93, 58, 48, 26, 23, 8, 9, 230, 190, 133, 108, 70, 56, 30, 26, 9, 10, 356, 260, 217, 148, 124, 76, 64, 34, 29, 10, 11

Offset: 1

Omar E. Pol, Feb 28 2012

Essentially this sequence is related to A206561 in the same way as A206563 is related to A181187. See the calculation in the example section of A206563.

			Triangle begins:
1;
2,   2;
7,   2,  3;
10, 10,  3,  4;
23, 12, 11,  4,  5;
36, 30, 17, 14,  5,  6;

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

Column 1-2: A066967, A066966. Right border is A000027.

Cf. A138785, A181187, A206561, A206563, A208476.

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local f, g;
      if n=0 then [1]
    elif i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));
         p (p (f, g), [0$i, g[1]])
      fi
    end:
T:= proc(n) local l;
      l:= b(n, n);
      seq (add (l[i+2*j+1]*(i+2*j), j=0..(n-i)/2), i=1..n)
    end:
seq (T(n), n=1..14);  # Alois P. Heinz, Mar 21 2012

p[f_, g_] := With[{m = Max[Length[f], Length[g]]}, PadRight[f, m, 0] + PadRight[g, m, 0]]; b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1}, i == 1, {1, n}, True, f = b[n, i-1]; g = If[i>n, {0}, b[n-i, i]]; p[p[f, g], Append[Array[0&, i], g[[1]]]]]]; T[n_] := Module[{l}, l = b[n, n]; Table[Sum[l[[i+2j+1]]*(i+2j), {j, 0, (n-i)/2}], {i, 1, n}]]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Mar 11 2015, after Alois P. Heinz *)

More terms from Alois P. Heinz, Mar 21 2012

A208476 Triangle read by rows: T(n,k) = total sum of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

Original entry on oeis.org

1, 1, 2, 5, 0, 3, 3, 8, 0, 4, 13, 2, 8, 0, 5, 13, 18, 6, 10, 0, 6

Offset: 1

Omar E. Pol, Feb 28 2012

Essentially this sequence is related to A206562 in the same way as A207032 is related to A207031 and also in the same way as A206563 is related to A181187. See the calculation in the example section of A206563.

			Triangle begins:
1;
1,   2;
5,   0,  3;
3,   8,  0,  4;
13,  2,  8,  0,  5;
13, 18,  6, 10,  0,  6;

Right border is A000027.

Cf. A135010, A182703, A206562, A207031, A207032, A207383, A208475.

A330372 Irregular triangle read by rows in which row n lists the self-conjugate partitions of n, ordered by their k-th largest parts, or 0 if such partitions does not exist.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 2, 3, 1, 1, 3, 2, 1, 4, 1, 1, 1, 4, 2, 1, 1, 3, 3, 2, 5, 1, 1, 1, 1, 3, 3, 3, 5, 2, 1, 1, 1, 4, 3, 2, 1, 6, 1, 1, 1, 1, 1, 4, 3, 3, 1, 6, 2, 1, 1, 1, 1, 5, 3, 2, 1, 1, 4, 4, 2, 2, 7, 1, 1, 1, 1, 1, 1, 5, 3, 3, 1, 1, 4, 4, 3, 2

Offset: 0

Omar E. Pol, Dec 17 2019

Row n lists the partitions of n whose Ferrers diagrams are symmetrics.
The k-th part of a partition equals the number of parts >= k of its conjugate partition. Hence, the k-th part of a self-conjugate partition equals the number of parts >= k.
The k-th rank of a partition is the k-th part minus the number of parts >= k. Thus all ranks of a conjugate-partitions are zero. Therefore row n lists the partitions of n whose n ranks are zero, n >= 1. For more information about the k-th ranks see A208478.

			Triangle begins (rows n = 0..10):
[0];
[1];
[0];
[2, 1];
[2, 2];
[3, 1, 1];
[3, 2, 1];
[4, 1, 1, 1];
[4, 2, 1, 1], [3, 3, 2];
[5, 1, 1, 1, 1], [3, 3, 3];
[5, 2, 1, 1, 1], [4, 3, 2, 1];
...
For n = 10 there are only two partitions of 10 whose Ferrers diagram are symmetric, they are [5, 2, 1, 1, 1] and [4, 3, 2, 1] as shown below:
  * * * * *
  * *
  *
  *
  *
            * * * *
            * * *
            * *
            *
So these partitions form the 10th row of triangle.
On the other hand, only two partitions of 10 have all their ranks equal to zero, they are [5, 2, 1, 1, 1] and [4, 3, 2, 1], so these partitions form the 10th row of triangle.

Freddy Barrera, Rows n = 0..50, flattened

Row n contains A000700(n) partitions.
The number of positive terms in row n is A067619(n).
Row sums give A330373.
Column 2 gives A000034.
Column 3 gives A000012.
For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483, A330370.

More terms from Freddy Barrera, Dec 31 2019

A207033 Total number of parts >= 3 in all partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 13, 22, 35, 54, 80, 121, 172, 247, 347, 484, 661, 906, 1215, 1632, 2162, 2855, 3730, 4871, 6290, 8111, 10381, 13252, 16802, 21269, 26750, 33583, 41948, 52277, 64862, 80326, 99055, 121922, 149541, 183052, 223350, 272038, 330343, 400450, 484154

Offset: 1

Omar E. Pol, Feb 18 2012

nonn

			a(4) = 2, because 2 parts have size >= 3 in all partitions of 4: [1,1,1,1], [1,1,2], [2,2], [1,3], [4].

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

Column 3 of A181187.

Cf. A006128, A096541, A206563.

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] +`if`(i>2, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[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]] + If[i > 2, g[[1]], 0]}]];
a[n_] := b[n, n][[2]];
Array[a, 50] (* Jean-François Alcover, Nov 12 2020, after Alois P. Heinz *)

G.f.: Sum_{k>=1} x^(3*k)/(1 - x^k) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

More terms from Alois P. Heinz, Feb 18 2012

A330371 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by the lower value of their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = n..1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 15 2019

In this triangle the partitions of n are ordered by their n-th rank. The partitions that have the same n-th rank appears ordered by their (n-1)-st rank. The partitions that have the same n-th rank and the same (n-1)-st rank appears ordered by their (n-2)-nd rank, and so on. The partitions that have all k-ranks equal zero appears ordered by their largest parts, then by their second largest parts, then by their third largest parts, and so on.
Note that a partition and its conjugate partition both are equidistants from the center of the list of partitions of n.
For further information see A330370.
First differs from A036037, A181317, A330370 and A334439 at a(48).
First differs from A080577 at a(56).

			Triangle begins:
[1];
[2], [1,1];
[3], [2,1], [1,1,1];
[4], [3,1], [2,2], [2,1,1], [1,1,1,1];
[5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1];
[6], [5,1], [4,2], [4,1,1], [3,3], [3,2,1], [2,2,2], [3,1,1,1], [2,2,1,1], ...
.
For n = 9 the 9th row of the triangle contains the partitions ordered as shown below:
---------------------------------------------------------------------------------
                                                                Ranks
          Conjugate
Label     with label    Partition                 k = 1  2  3  4  5  6  7  8  9
---------------------------------------------------------------------------------
   1         30         [9]                           8 -1 -1 -1 -1 -1 -1 -1 -1
   2         29         [8, 1]                        6  0 -1 -1 -1 -1 -1 -1  0
   3         28         [7, 2]                        5  0 -1 -1 -1 -1 -1  0  0
   4         27         [7, 1, 1]                     4  0  0 -1 -1 -1 -1  0  0
   5         26         [6, 3]                        4  1 -2 -1 -1 -1  0  0  0
   6         25         [6, 2, 1]                     3  0  0 -1 -1 -1  0  0  0
   7         24         [6, 1, 1, 1]                  2  0  0  0 -1 -1  0  0  0
   8         23         [5, 4]                        3  2 -2 -2 -1  0  0  0  0
   9         22         [5, 3, 1]                     2  1 -1 -1 -1  0  0  0  0
  10         21         [5, 2, 2]                     2 -1  1 -1 -1  0  0  0  0
  11         20         [5, 2, 1, 1]                  1  0  0  0 -1  0  0  0  0
  12         19         [4, 4, 1]                     1  2 -1 -2  0  0  0  0  0
  13         18         [4, 3, 2]                     1  0  0 -1  0  0  0  0  0
  14         17         [4, 3, 1, 1]                  0  1 -1  0  0  0  0  0  0
  15  (self-conjugate)  [5, 1, 1, 1, 1]  All zeros -> 0  0  0  0  0  0  0  0  0
  16  (self-conjugate)  [3, 3, 3]        All zeros -> 0  0  0  0  0  0  0  0  0
  17         14         [4, 2, 2, 1]                  0 -1  1  0  0  0  0  0  0
  18         13         [3, 3, 2, 1]                 -1  0  0  1  0  0  0  0  0
  19         12         [3, 2, 2, 2]                 -1 -2  1  2  0  0  0  0  0
  20         11         [4, 2, 1, 1, 1]              -1  0  0  0  1  0  0  0  0
  21         10         [3, 3, 1, 1, 1]              -2  1 -1  1  1  0  0  0  0
  22          9         [3, 2, 2, 1, 1]              -2 -1  1  1  1  0  0  0  0
  23          8         [2, 2, 2, 2, 1]              -3 -2  2  2  1  0  0  0  0
  24          7         [4, 1, 1, 1, 1, 1]           -2  0  0  0  1  1  0  0  0
  25          6         [3, 2, 1, 1, 1, 1]           -3  0  0  1  1  1  0  0  0
  26          5         [2, 2, 2, 1, 1, 1]           -4 -1  2  1  1  1  0  0  0
  27          4         [3, 1, 1, 1, 1, 1, 1]        -4  0  0  1  1  1  1  0  0
  28          3         [2, 2, 1, 1, 1, 1, 1]        -5  0  1  1  1  1  1  0  0
  29          2         [2, 1, 1, 1, 1, 1, 1, 1]     -6  0  1  1  1  1  1  1  0
  30          1         [1, 1, 1, 1, 1, 1, 1, 1, 1]  -8  1  1  1  1  1  1  1  1

Another version of A330370.
Row n contains A000041(n) partitions.
Row n has length A006128(n).
The sum of n-th row is A066186(n).
Cf. A000700, A036037, A080577, A181317, A207031, A330368, A330370, A334439.
For the "k-th rank" see also: A181187, A208478, A208479, A208482, A208483.

cp's OEIS Frontend

A208475 Triangle read by rows: T(n,k) = total sum of odd/even parts >= k in all partitions of n, if k is odd/even.

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A208476 Triangle read by rows: T(n,k) = total sum of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

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A330372 Irregular triangle read by rows in which row n lists the self-conjugate partitions of n, ordered by their k-th largest parts, or 0 if such partitions does not exist.

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A207033 Total number of parts >= 3 in all partitions of n.

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A330371 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by the lower value of their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = n..1.

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