cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A211985 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as a spiral.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Aug 19 2012

Keywords

Comments

In order to construct this sequence we use the following rules:
- Consider the partitions of positive integers.
- For each positive integer its shells must be arranged in a spiral.
- The sequence lists one spiral for each positive integer.
- If the integer j is odd then we use the same spiral of A211995.
- If the integer j is even then the first composition listed of each spiral is j.

Examples

			--------------------------------------------
.               Expanded        Geometric
Compositions   arrangement        model
--------------------------------------------
1;                 1;             |*|
--------------------------------------------
2;                 . 2;           |* *|
1,1;               1,1;           |o|*|
--------------------------------------------
3;               3 . .;         |* * *|
1,1,1;           1,1,1;         |*|o|o|
1,2;             1,. 2;         |*|o o|
--------------------------------------------
4,;              . . . 4;       |* * * *|
2,2;             . 2,. 2;       |* *|* *|
1,2,1;           1,. 2,1;       |o|o o|*|
1,1,1,1,;        1,1,1,1;       |o|o|o|*|
3,1;             3 . .,1;       |o o o|*|
--------------------------------------------
5;             5 . . . .;     |* * * * *|
2,3;           2 .,3 . .;     |* *|* * *|
1,3,1;         1,3 . .,1;     |*|o o o|o|
1,1,1,1,1;     1,1,1,1,1;     |*|o|o|o|o|
1,1,2,1;       1,1,. 2,1;     |*|o|o o|o|
1,2,2;         1,. 2,. 2;     |*|o o|o o|
1,4;           1,. . . 4;     |*|o o o o|
--------------------------------------------
6;             . . . . . 6;   |* * * * * *|
3,3;           . . 3,. . 3;   |* * *|* * *|
4,2;           . . . 4,. 2;   |* * * *|* *|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
1,4,1;         1,. . . 4,1;   |o|o o o o|*|
1,2,2,1;       1,. 2,. 2,1;   |o|o o|o o|*|
1,1,2,1,1;     1,1,. 2,1,1;   |o|o|o o|o|*|
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
1,3,1,1;       1,3 . .,1,1;   |o|o o o|o|*|
2,3,1;         2 .,3 . .,1;   |o o|o o o|*|
5,1;           5 . . . .,1;   |o o o o o|*|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Crossrefs

Rows sums give A036042, n>=1.
Mirror of A211986. Other spiral versions are A211987, A211988, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211989 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Aug 18 2012

Keywords

Comments

The sequence lists the partitions of all positive integers. Each row of the irregular array is a partition of j.
At stage 1, we start with 1.
At stage j > 1, we write the partitions of j using the following rules:
First, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, starting with the partition that contains the part of size j.
Second, we copy from this array the partitions of j-1 in descending order, as a mirror image, starting with the partition that contains the part of size j-2 together with the part of size 1. At the end of each new row, we added a part of size 1.

Examples

			A table of partitions.
--------------------------------------------
.              Expanded       Geometric
Partitions     arrangement    model
--------------------------------------------
1;             1;             |*|
--------------------------------------------
2;             . 2;           |* *|
1,1;           1,1;           |o|*|
--------------------------------------------
3;             . . 3;         |* * *|
1,1,1;         1,1,1;         |o|o|*|
2,1;           . 2,1;         |o o|*|
--------------------------------------------
4;             . . . 4;       |* * * *|
2,2;           . 2,. 2;       |* *|* *|
2,1,1;         . 2,1,1;       |o o|o|*|
1,1,1,1;       1,1,1,1;       |o|o|o|*|
3,1;           . . 3,1;       |o o o|*|
--------------------------------------------
5;             . . . . 5;     |* * * * *|
3,2;           . . 3,. 2;     |* * *|* *|
3,1,1;         . . 3,1,1;     |o o o|o|*|
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|
2,2,1;         . 2,. 2,1;     |o o|o o|*|
4,1;           . . . 4,1;     |o o o o|*|
--------------------------------------------
6;             . . . . . 6;   |* * * * * *|
3,3;           . . 3,. . 3;   |* * *|* * *|
4,2;           . . . 4,. 2;   |* * * *|* *|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
4,1,1;         . . . 4,1,1;   |o o o o|o|*|
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|
5,1;           . . . . 5,1;   |o o o o o|*|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Crossrefs

Rows sums give A036042, n>=1.
Other versions are A211983, A211984, A211999. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211994 A list of ordered partitions of the positive integers.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Aug 18 2012

Keywords

Comments

The order of the partitions of the odd integers is the same as A026792. The order of the partitions of the even integers is the same as A211992.

Examples

			A table of partitions.
--------------------------------------------
.              Expanded       Geometric
Partitions     arrangement    model
--------------------------------------------
1;             1;             |*|
--------------------------------------------
1,1;           1,1;           |o|*|
2;             . 2;           |* *|
--------------------------------------------
3;             . . 3;         |* * *|
2,1;           . 2,1;         |o o|*|
1,1,1;         1,1,1;         |o|o|*|
--------------------------------------------
1,1,1,1;       1,1,1,1;       |o|o|o|*|
2,1,1;         . 2,1,1;       |o o|o|*|
3,1;           . . 3,1;       |o o o|*|
2,2;           . 2,. 2;       |* *|* *|
4;             . . . 4;       |* * * *|
--------------------------------------------
5;             . . . . 5;     |* * * * *|
3,2;           . . 3,. 2;     |* * *|* *|
4,1;           . . . 4,1;     |o o o o|*|
2,2,1;         . 2,. 2,1;     |o o|o o|*|
3,1,1;         . . 3,1,1;     |o o o|o|*|
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|
--------------------------------------------
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|
4,1,1;         . . . 4,1,1;   |o o o o|o|*|
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|
5,1;           . . . . 5,1;   |o o o o o|*|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
4,2;           . . . 4,. 2;   |* * * *|* *|
3,3;           . . 3,. . 3;   |* * *|* * *|
6;             . . . . . 6;   |* * * * * *|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Crossrefs

Rows sums give A036042, n>=1.
Other versions are A026792, A211992, A211993. See also A211983, A211984, A211989, A211999. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

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, 2, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 5, 2, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 6, 2, 5, 3, 4, 4, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 7, 2, 6, 3, 5, 4, 5, 2, 2, 4, 3, 2, 3, 3, 3, 3, 2, 2
Offset: 1

Views

Author

Omar E. Pol, Mar 21 2008

Keywords

Comments

The remainder of row n is necessarily A000041(n-1) 1's.
Previous name: A shell model of partitions. Row n lists the parts of the last section of the set of partitions of n.
Row n lists the nonzero terms of the n-th row of A138136 together with A000041(n-1) 1's.
Row n is also the n-th row of A138138 in reverse order.

Examples

			Triangle begins:
1
2,1
3,1,1
4,2,2,1,1,1
5,3,2,1,1,1,1,1,
6,4,2,3,3,2,2,2,1,1,1,1,1,1,1
7,5,2,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1
8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

Links

Crossrefs

Mirror of A138138.
Row lengths give A138137.
Row sums give A138879.
Column 1 gives A000027.
Right border gives A000012.
Another version is A138121 which is the mirror of A135010.
Cf. A000041, A080577, A138136, A139094, A139100.

Programs

  • Mathematica
    Table[Cases[IntegerPartitions[n], x_ /; Last[x] != 1] ~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 8}] // Flatten (* Robert Price, May 22 2020 *)

Extensions

New name and comments edited by Peter Munn and Omar E. Pol, Jul 25 2025

A340426 Triangle read by rows: T(n,k) = A000203(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 3, 0, 4, 0, 1, 7, 0, 3, 1, 6, 0, 4, 3, 2, 12, 0, 7, 4, 6, 2, 8, 0, 6, 7, 8, 6, 4, 15, 0, 12, 6, 14, 8, 12, 4, 13, 0, 8, 12, 12, 14, 16, 12, 7, 18, 0, 15, 8, 24, 12, 28, 16, 21, 8, 12, 0, 13, 15, 16, 24, 14, 28, 28, 24, 12, 28, 0, 18, 13, 30, 16, 48, 24, 49, 32, 36, 14, 14, 0, 12
Offset: 1

Views

Author

Omar E. Pol, Jan 07 2021

Keywords

Comments

Conjecture: the sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.

Examples

			Triangle begins:
   1;
   3,  0;
   4,  0,  1;
   7,  0,  3,  1;
   6,  0,  4,  3,  2;
  12,  0,  7,  4,  6,  2;
   8,  0,  6,  7,  8,  6,  4;
  15,  0, 12,  6, 14,  8, 12,  4;
  13,  0,  8, 12, 12, 14, 16, 12,  7;
  18,  0, 15,  8, 24, 12, 28, 16, 21,  8;
  12,  0, 13, 15, 16, 24, 14, 28, 28, 24, 12;
  28,  0, 18, 13, 30, 16, 48, 24, 49, 32, 36, 14;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *   12  =  12
2      0   *   6   =   0
3      1   *   7   =   7
4      1   *   4   =   4
5      2   *   3   =   6
6      2   *   1   =   2
.           A000203
--------------------------
The sum of row 6 is 12 + 0 + 7 + 4 + 6 + 2 = 31, equaling A138879(6) = 31.

Crossrefs

Columns 1, 3 and 4 give A000203.
Column 2 gives A000004.
Columns 5 and 6 gives A074400.
Column 7 and 8 give A239050.
Column 9 gives A319527.
Column 10 gives A319528.
Leading diagonal gives A002865.
Cf. A135010, A138879.

A194448 Number of parts > 1 in the n-th region of the shell model of partitions.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 14, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 21, 1, 2, 1, 4, 1, 2
Offset: 1

Views

Author

Omar E. Pol, Dec 10 2011

Keywords

Comments

Also triangle read by rows: T(n,k) = number of parts > 1 in the k-th region of the last section of the set of partitions of n.

Examples

			Written as a triangle:
0;
1;
1;
1,2;
1,2;
1,2,1,4;
1,2,1,4;
1,2,1,4,1,1,7;
1,2,1,4,1,2,1,8;
1,2,1,4,1,1,7,1,2,1,1,12;
1,2,1,4,1,2,1,8,1,1,3,1,1,14;
1,2,1,4,1,1,7,1,2,1,1,12,1,2,1,4,1,2,1,1,21;

Crossrefs

Row sums give A138135. Row n has length A187219(n).
Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447, A194449.

A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

Original entry on oeis.org

0, 1, 2, 5, 7, 16, 20, 39, 52, 86, 113, 184, 232, 353, 462, 661, 851, 1202, 1526, 2098, 2670, 3565, 4514, 5967, 7473, 9715, 12162, 15583, 19373, 24625, 30410, 38274, 47112, 58725, 71951, 89129, 108599, 133612, 162259, 198346, 239825, 291718, 351269, 425102
Offset: 1

Views

Author

Omar E. Pol, Feb 20 2012

Keywords

Examples

			For n = 7 the last section of the set of partitions of 7 looks like this:
.
.        (. . . . . . 7)
.        (. . . 4 . . 3)
.        (. . . . 5 . 2)
.        (. . 3 . 2 . 2)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.
The sum of all parts = 7+4+3+5+2+3+2+2+1*11 = 39, on the other hand the total number of parts is 1+2+2+3+1*11 = 19, so a(7) = 39 - 19 = 20. Note that the number of dots in the picture is also equal to a(7) = 6+5+5+4 = 20.

Links

Crossrefs

Row sums of triangle A207034. Partial sums give A196087.
Cf. A006128, A066186, A135010, A138121, A138135, A138137, A138879, A138880, A187219, A194548, A207038.

Programs

  • Maple
    b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<2 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]*(i-1)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);  # Alois P. Heinz, Feb 20 2012
  • Mathematica
    b[n_, i_] := b[n, i] = Module[{f, g}, Which[n==0, {1, 0}, i<2, {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]]*(i-1)}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 13 2015, after Alois P. Heinz *)

Formula

a(n) = A138879(n) - A138137(n) = A138880(n) - A138135(n). - Omar E. Pol, Apr 21 2012
G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^2 / Product_{j>=2} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

Extensions

More terms from Alois P. Heinz, Feb 20 2012

A206435 Total sum of odd parts in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 5, 3, 13, 13, 29, 29, 66, 70, 126, 146, 241, 287, 450, 526, 791, 963, 1360, 1660, 2312, 2810, 3799, 4649, 6158, 7528, 9824, 11962, 15393, 18773, 23804, 28932, 36413, 44093, 54953, 66419, 82085, 98929, 121469, 145865, 177983, 213241, 258585, 308861
Offset: 1

Views

Author

Omar E. Pol, Feb 12 2012

Keywords

Comments

From Omar E. Pol, Apr 09 2023: (Start)
Convolution of A002865 and A000593.
a(n) is also the total sum of odd divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the sum of odd terms in the n-th row of the triangle A207378.
a(n) is also the sum of odd terms in the n-th row of the triangle A336812. (End)

Links

Crossrefs

Partial sums give A066967.
Cf. A000593, A002865, A135010, A138121, A138879, A206433, A206434, A206436, A207378, A336811, A336812.

Programs

  • Maple
    b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<1 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] +(i mod 2)*h[1]*i]
      fi
    end:
a:= n-> b(n, n)[2] -`if`(n=1, 0, b(n-1, n-1)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 16 2012
  • Mathematica
    b[n_, i_] := b[n, i] = Module[{g, h}, Which[n == 0, {1, 0}, i < 1, {0, 0}, True, g = b[n, i-1]; h = If[i > n, {0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + Mod[i, 2]*h[[1]]*i}]]; a[n_] := b[n, n][[2]] - If[n == 1, 0, b[n-1, n-1][[2]]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

Formula

G.f.: (Sum_{i>=0} (2*i+1)*x^(2*i)*(1-x)/(1-x^(2*i+1))) / Product_{j>0} (1-x^j). - Alois P. Heinz, Mar 16 2012
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2*n)). - Vaclav Kotesovec, May 29 2018

Extensions

More terms from Alois P. Heinz, Mar 16 2012

A206436 Total sum of even parts in the last section of the set of partitions of n.

Original entry on oeis.org

0, 2, 0, 8, 2, 18, 10, 42, 28, 80, 70, 162, 148, 290, 300, 530, 562, 918, 1020, 1570, 1780, 2602, 3022, 4286, 4992, 6858, 8110, 10872, 12888, 16962, 20178, 26134, 31138, 39728, 47412, 59848, 71312, 89072, 106176, 131440, 156400, 192164, 228330, 278616, 330502
Offset: 1

Views

Author

Omar E. Pol, Feb 12 2012

Keywords

Comments

Also total sum of even parts in the partitions of n that do not contain 1 as a part.
From Omar E. Pol, Apr 09 2023: (Start)
Convolution of A002865 and A146076.
a(n) is also the total sum of even divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the sum of even terms in the n-th row of the triangle A207378.
a(n) is also the sum of even terms in the n-th row of the triangle A336812. (End)

Links

Crossrefs

Partial sums give A066966.
Cf. A002865, A135010, A138121, A138879, A146076, A206433, A206434, A206435, A207378, A336811, A336812.

Programs

  • Maple
    b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<1 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] +((i+1) mod 2)*h[1]*i]
      fi
    end:
a:= n-> b(n, n)[2] -`if`(n=1, 0, b(n-1, n-1)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 16 2012
  • Mathematica
    b[n_, i_] := b[n, i] = Module[{g, h}, Which[n == 0, {1, 0}, i < 1, {0, 0}, True, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + Mod[i+1, 2]*h[[1]]*i}]]; a[n_] := b[n, n][[2]] - If[n == 1, 0, b[n-1, n-1][[2]]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

Formula

G.f.: (Sum_{i>0} 2*i*x^(2*i)*(1-x)/(1-x^(2*i))) / Product_{i>0} (1-x^i). - Alois P. Heinz, Mar 16 2012
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2*n)). - Vaclav Kotesovec, May 29 2018

Extensions

More terms from Alois P. Heinz, Mar 16 2012

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

Views

Author

Omar E. Pol, Feb 15 2012

Keywords

Examples

			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;

Links

Crossrefs

Columns 1-2 give A138879, A138880. Diagonal is A000027.
Cf. A135010, A138121, A182703, A206561, A207031, A207032.
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