A236626 -id:A236626 - cp's OEIS frontend

A236625 Total number of parts in all overcompositions of n.

0, 2, 6, 24, 66, 180, 496, 1272, 3202, 7798, 18980, 45076, 106288, 246956, 568776, 1299184, 2944654, 6630660, 14838606, 33026000, 73126376, 161198136, 353812612, 773645124, 1685548792, 3660364490, 7924414752, 17107225340, 36832846344, 79107019964, 169505684844

Offset: 0

Omar E. Pol, Feb 01 2014

nonn

For the definition of overcomposition see A236002.
The equivalent sequence for overpartitions is A235792.
Row sums of triangle A236628.

			For n = 3 the 12 overcompositions of 3 are [3], [3'], [1, 2], [1', 2], [1, 2'], [1', 2'], [2, 1], [2', 1], [2, 1'], [2', 1'], [1, 1, 1], [1', 1, 1]. There are 24 parts, so a(3) = 24.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A001792, A006128, A235999, A236002, A235792, A236626, A236628, A236633.

b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
      `if`(i<1, 0, add((p-> p+[0, p[1]*j])(1/j!*
      `if`(j>0, 2, 1)*b(n-i*j, i-1, p+j)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..35);  # Alois P. Heinz, Apr 28 2016

b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0}, If[i < 1, {0, 0}, Sum[# + {0, #[[1]]*j}&[1/j!*If[j > 0, 2, 1]*b[n - i*j, i - 1, p + j]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 03 2022, after Alois P. Heinz *)

a(6)-a(30) from Alois P. Heinz, Feb 02 2014

A236628 Triangle read by rows in which T(n,k) is the number of parts in the k-th region of the set of overcompositions of n, with overcompositions in colexicographic order.

Original entry on oeis.org

2, 2, 4, 2, 6, 4, 12, 2, 6, 4, 14, 4, 6, 4, 26

Offset: 1

Omar E. Pol, Feb 01 2014

Right border gives A236002, n >= 1.

Row sums give A236625, n >= 1.

			Written as an irregular triangle in which row n has length 2^n the sequence begins:
2;
2, 4;
2, 6, 4, 12;
2, 6, 4, 14, 4, 6, 4, 26;
...
For n = 3 the diagram shows the four regions of the overcompositions of 3, with overcompositions in colexicographic order.
------------------------------------------------
.  Diagram of            Regions of the diagram
overcompositions        ------------------------
.    of 3                k: 1     2   3       4
------------------------------------------------
.    _ _ _                  _     _           _
1   |.| | |                |.|   | |         | |
2   |_| | |                |_|  _| |         | |
3   |  .|.|                    |  .|         |.|
4   |   |.|                    |   |         |.|
5   |  .| |                    |  .|         | |
6   |_ _| |                    |_ _|  _     _| |
7   |.|  .|                          |.|   |  .|
8   | |  .|                          | |   |  .|
9   |.|   |                          |.|   |   |
10  |_|   |                          |_|  _|   |
11  |    .|                              |    .|
12  |_ _ _|                              |_ _ _|
...
Number of parts.........:   2     6   4      12
.
Every row of every region contains only one part.

Cf. A236002, A236625, A236626.

cp's OEIS Frontend

A236625 Total number of parts in all overcompositions of n.

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