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 1-7 of 7 results.

A236633 Number of overcompositions of n minus the number of compositions of n.

Original entry on oeis.org

0, 1, 2, 8, 18, 44, 112, 260, 600, 1346, 3064, 6784, 15020, 32812, 71328, 154320, 332026, 711500, 1518384, 3229044, 6843256, 14464760, 30487496, 64112960, 134515472, 281671698, 588680628, 1228211140, 2558366188, 5321151540, 11052034932, 22925310868
Offset: 0

Views

Author

Omar E. Pol, Feb 02 2014

Keywords

Comments

Number of overcompositions of n that contain at least one overlined part.

Examples

			For n = 3 the number of overcompositions of 3 is A236002(3) = 12 and the number of compositions of 3 is A011782(3) = 4, so a(3) = 12 - 4 = 8.
On the other hand, 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 8 overcompositions with at least one overlined part, so a(3) = 8.

Crossrefs

Cf. A011782, A236002, A230441.

Formula

a(n) = A236002(n) - A011782(n).

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).

A335651 a(n) is the sum, over all overpartitions of n, of the non-overlined parts.

Original entry on oeis.org

1, 5, 14, 35, 74, 150, 280, 505, 875, 1470, 2402, 3850, 6034, 9300, 14120, 21131, 31220, 45619, 65930, 94374, 133892, 188350, 262904, 364350, 501459, 685762, 932200, 1259944, 1693750, 2265380, 3015152, 3994585, 5268988, 6920700, 9053748, 11798873, 15319610, 19820738, 25557560
Offset: 1

Views

Author

Jeremy Lovejoy, Jun 17 2020

Keywords

Examples

			The 8 overpartitions of 3 are [3], [3'], [2,1], [2,1'], [2',1], [2',1'], [1,1,1], [1',1,1], and so a(3) = 14.

Links

Crossrefs

Cf. A015128, A230441, A235790, A235792, A235793, A235798, A236000, A236001, A237047, A335666.
Cf. A305102 (number of non-overlined parts).

Programs

  • PARI
    my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, k*q^k/(1-q^k)) ) \\ Joerg Arndt, Jun 18 2020

Formula

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) * Sum_{n>=1} n*q^n/(1-q^n).
a(n) = A235793(n) - A335666(n). - Omar E. Pol, Jun 17 2020

A335666 a(n) is the sum, over all overpartitions of n, of the overlined parts.

Original entry on oeis.org

1, 3, 10, 21, 46, 90, 168, 295, 511, 850, 1382, 2198, 3430, 5260, 7960, 11861, 17468, 25445, 36670, 52346, 74092, 103986, 144840, 200322, 275191, 375662, 509816, 687960, 923442, 1233340, 1639312, 2168999, 2857460, 3748772, 4898652, 6377023, 8271294, 10690830, 13771912, 17683642
Offset: 1

Views

Author

Jeremy Lovejoy, Jun 17 2020

Keywords

Examples

			The 8 overpartitions of 8 are [3], [3'], [2,1], [2,1'], [2',1], [2',1'], [1,1,1], [1',1,1], and so a(3) = 10.

Links

Crossrefs

Cf. A015128, A230441, A235790, A235792, A235793, A235798, A236000, A236001, A237047, A335651.
Cf. A305101 (number of overlined parts).

Programs

  • PARI
    my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, k*q^k/(1+q^k)) ) \\ Joerg Arndt, Jun 18 2020

Formula

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) * Sum_{n>=1} n*q^n/(1+q^n).
a(n) = A235793(n) - A335651(n). - Omar E. Pol, Jun 17 2020

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 1-7 of 7 results.

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