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.

Showing 1-6 of 6 results.

A236625 Total number of parts in all overcompositions of n.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Feb 01 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Extensions

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

A237044 Number of overcompositions of n minus the number of partitions of n.

Original entry on oeis.org

0, 1, 2, 9, 21, 53, 133, 309, 706, 1572, 3534, 7752, 16991, 36807, 79385, 170528, 364563, 776739, 1649071, 3490698, 7366917, 15512544, 32583646, 68306009, 142902505, 298446956, 622232624, 1295316994, 2692580198, 5589582431, 11588900240, 23999045850
Offset: 0

Views

Author

Omar E. Pol, Feb 02 2014

Keywords

Crossrefs

Cf. A000041, A011782, A015128, A056823, A230441, A236002, A236633, A237045, A235047, A237272.

Formula

a(n) = A236002(n) - A000041(n).

A237045 Number of overcompositions of n minus the number of overpartitions of n.

Original entry on oeis.org

0, 0, 0, 4, 12, 36, 104, 260, 628, 1448, 3344, 7464, 16564, 36180, 78480, 169232, 362732, 774172, 1645508, 3485788, 7360208, 15503432, 32571360, 68289536, 142880552, 298417848, 622194236, 1295266596, 2692514348, 5589496748, 11588789220, 23998902548
Offset: 0

Views

Author

Omar E. Pol, Feb 02 2014

Keywords

Comments

Number of overcompositions of n that contain at least two parts in increasing order.

Examples

			Illustration of a(4) = -6 with both overcompositions and overpartitions in colexicographic order.
--------------------------------------------------------
.    Overcompositions of 4      Overpartitions of 4
--------------------------------------------------------
.    _ _ _ _                    _ _ _ _
1   |.| | | |  1', 1,  1,  1   |.| | | |  1', 1,  1,  1
2   |_| | | |  1,  1,  1,  1   |_| | | |  1,  1,  1,  1
3   |  .|.| |  2', 1', 1       |  .|.| |  2', 1', 1
4   |   |.| |  2,  1', 1       |   |.| |  2,  1', 1
5   |  .| | |  2', 1,  1       |  .| | |  2', 1,  1
6   |_ _| | |  2,  1,  1       |_ _| | |  2,  1,  1
7  *|.|  .| |  1', 2', 1       |    .|.|  3', 1
8  *| |  .| |  1,  2', 1       |     |.|  3,  1
9  *|.|   | |  1', 2,  1       |    .| |  3', 1
10 *|_|   | |  1,  2,  1       |_ _ _| |  3,  1
11  |    .|.|  3', 1'          |  .|   |  2', 2
12  |     |.|  3,  1'          |_ _|   |  2,  2
13  |    .| |  3', 1           |      .|  4'
14  |_ _ _| |  3,  1           |_ _ _ _|  4
15 *|.| |  .|  1', 1,  2'
16 *| | |  .|  1,  1,  2'
17 *|.| |   |  1', 1,  2
18 *|_| |   |  1,  1,  2
19  |  .|   |  2', 2
20  |_ _|   |  2,  2
21 *|.|    .|  1', 3'
22 *| |    .|  1,  3'
23 *|.|     |  1', 3
24 *|_|     |  1,  3
25  |      .|  4'
26  |_ _ _ _|  4
.
There are 26 overcompositions of 4 and there are 14 overpartitions of 4, so the difference is a(4) = 26 - 14 = 12.
On the other hand there are 12 overcompositions of 4 that contain at least two parts in increasing order, so a(4) = 12.

Crossrefs

Cf. A000041, A011782, A015128, A056823, A230441, A236002, A236633, A237044, A237047, A237272.

Formula

a(n) = A236002(n) - A015128(n).

A237047 Number of compositions of n minus the number of overpartitions of n.

Original entry on oeis.org

0, -1, -2, -4, -6, -8, -8, 0, 28, 102, 280, 680, 1544, 3368, 7152, 14912, 30706, 62672, 127124, 256744, 516952, 1038672, 2083864, 4176576, 8365080, 16746150, 33513608, 67055456, 134148160, 268345208, 536754288, 1073591680, 2147291036, 4294721040, 8589620784
Offset: 0

Views

Author

Omar E. Pol, Feb 02 2014

Keywords

Comments

Note that a(7) = 0 therefore 7 is the only positive integer whose number of compositions equals the number of overpartitions: A011782(7) = A015128(7) = 64.

Examples

			Illustration of a(4) = -6.
--------------------------------------------------------
.     Compositions of 4          Overpartitions of 4
--------------------------------------------------------
.    _ _ _ _                    _ _ _ _
1   |_| | | |  1, 1, 1, 1      |.| | | |  1', 1,  1,  1
2   |_ _| | |  2, 1, 1         |_| | | |  1,  1,  1,  1
3   |_|   | |  1, 2, 1         |  .|.| |  2', 1', 1
4   |_ _ _| |  3, 1            |   |.| |  2,  1', 1
5   |_| |   |  1, 1, 2         |  .| | |  2', 1,  1
6   |_ _|   |  2, 2            |_ _| | |  2,  1,  1
7   |_|     |  1, 3            |    .|.|  3', 1
8   |_ _ _ _|  4               |     |.|  3,  1
9                              |    .| |  3', 1
10                             |_ _ _| |  3,  1
11                             |  .|   |  2', 2
12                             |_ _|   |  2,  2
13                             |      .|  4'
14                             |_ _ _ _|  4
.
There are 8 compositions of 4 and there are 14 overpartitions of 4, so a(4) = 8 - 14 = -6.

Crossrefs

Cf. A011782, A015128, A056823, A230441, A236633, A237044, A237045.

Formula

a(n) = A011782(n) - A015128(n).

A236626 Sum of all parts of all overcompositions of n.

Original entry on oeis.org

2, 8, 36, 104, 300, 864, 2268, 5824, 14418, 35760, 85888, 204816, 479804, 1113280, 2560560, 5836704, 13209612, 29690208, 66332572, 147350880, 325780056, 716862256, 1571067072, 3429697920, 7461222850, 16178111560, 34973640108, 75392349648
Offset: 1

Views

Author

Omar E. Pol, Feb 01 2014

Keywords

Comments

For the definition of overcomposition see A236002.
The equivalent sequence for overpartitions is A235793.

Examples

			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], hence the sum of all parts is 3+3+1+2+1+2+1+2+1+2+2+1+2+1+2+1+2+1+1+1+1+1+1+1 = 3*12 = 36, so a(3) = 36.

Crossrefs

Cf. A001787, A066186, A235793, A235999, A236002, A236625, A236628, A236633.

Formula

a(n) = n*A236002(n).

A237272 Number of overcompositions of n that contain at least two parts in increasing order and that contain at least one overlined part.

Original entry on oeis.org

0, 0, 0, 3, 9, 27, 83, 211, 522, 1222, 2874, 6496, 14593, 32185, 70423, 153024, 330195, 708933, 1514821, 3224134, 6836547, 14455648, 30475210, 64096487, 134493519, 281642590, 588642240, 1228160742, 2558300338, 5321065857, 11051923912, 22925167566
Offset: 0

Views

Author

Omar E. Pol, Feb 09 2014

Keywords

Comments

Number of overcompositions of n minus the number of overpartitions of n plus the number of partitions of n minus the number of compositions of n.

Crossrefs

Cf. A000041, A011782, A015128, A056823, A230441, A236002, A236633, A237044, A237045.

Formula

a(n) = A236002(n) - A015128(n) + A000041(n) - A011782(n) = A236002(n) - A230441(n) - A011782(n) = A237045(n) - A056823(n).
Showing 1-6 of 6 results.

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