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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A138121 Triangle read by rows in which row n lists the partitions of n that do not contain 1 as a part in juxtaposed reverse-lexicographical order followed by A000041(n-1) 1's.

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

Views

Author

Omar E. Pol, Mar 21 2008

Keywords

Comments

Mirror of triangle A135010.

Examples

			Triangle begins:
[1];
[2],[1];
[3],[1],[1];
[4],[2,2],[1],[1],[1];
[5],[3,2],[1],[1],[1],[1],[1];
[6],[3,3],[4,2],[2,2,2],[1],[1],[1],[1],[1],[1],[1];
[7],[4,3],[5,2],[3,2,2],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1];
...
The illustration of the three views of the section model of partitions (version "tree" with seven sections) shows the connection between several sequences.
---------------------------------------------------------
Partitions                A194805            Table 1.0
.  of 7       p(n)        A194551             A135010
---------------------------------------------------------
7              15                    7     7 . . . . . .
4+3                                4       4 . . . 3 . .
5+2                              5         5 . . . . 2 .
3+2+2                          3           3 . . 2 . 2 .
6+1            11    6       1             6 . . . . . 1
3+3+1                  3     1             3 . . 3 . . 1
4+2+1                    4   1             4 . . . 2 . 1
2+2+2+1                    2 1             2 . 2 . 2 . 1
5+1+1           7            1   5         5 . . . . 1 1
3+2+1+1                      1 3           3 . . 2 . 1 1
4+1+1+1         5        4   1             4 . . . 1 1 1
2+2+1+1+1                  2 1             2 . 2 . 1 1 1
3+1+1+1+1       3            1 3           3 . . 1 1 1 1
2+1+1+1+1+1     2          2 1             2 . 1 1 1 1 1
1+1+1+1+1+1+1   1            1             1 1 1 1 1 1 1
.               1                         ---------------
.               *<------- A000041 -------> 1 1 2 3 5 7 11
.                         A182712 ------->   1 0 2 1 4 3
.                         A182713 ------->     1 0 1 2 2
.                         A182714 ------->       1 0 1 1
.                                                  1 0 1
.                         A141285           A182703  1 0
.                    A182730   A182731                 1
---------------------------------------------------------
.                              A138137 --> 1 2 3 6 9 15..
---------------------------------------------------------
.       A182746 <--- 4 . 2 1 0 1 2 . 4 ---> A182747
---------------------------------------------------------
.
.       A182732 <--- 6 3 4 2 1 3 5 4 7 ---> A182733
.                    . . . . 1 . . . .
.                    . . . 2 1 . . . .
.                    . 3 . . 1 2 . . .
.      Table 2.0     . . 2 2 1 . . 3 .     Table 2.1
.                    . . . . 1 2 2 . .
.                            1 . . . .
.
.  A182982  A182742       A194803       A182983  A182743
.  A182992  A182994       A194804       A182993  A182995
---------------------------------------------------------
.
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms (n = 1..6). The table shows the six sections of the set of partitions of 6. Note that before the dissection the set of partitions was in the ordering mentioned in A026792. More generally, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
Illustration of initial terms:
---------------------------------------
n  j     Diagram          Parts
---------------------------------------
.         _
1  1     |_|              1;
.         _ _
2  1     |_  |            2,
2  2       |_|            .  1;
.         _ _ _
3  1     |_ _  |          3,
3  2         | |          .  1,
3  3         |_|          .  .  1;
.         _ _ _ _
4  1     |_ _    |        4,
4  2     |_ _|_  |        2, 2,
4  3           | |        .  1,
4  4           | |        .  .  1,
4  5           |_|        .  .  .  1;
.         _ _ _ _ _
5  1     |_ _ _    |      5,
5  2     |_ _ _|_  |      3, 2,
5  3             | |      .  1,
5  4             | |      .  .  1,
5  5             | |      .  .  1,
5  6             | |      .  .  .  1,
5  7             |_|      .  .  .  .  1;
.         _ _ _ _ _ _
6  1     |_ _ _      |    6,
6  2     |_ _ _|_    |    3, 3,
6  3     |_ _    |   |    4, 2,
6  4     |_ _|_ _|_  |    2, 2, 2,
6  5               | |    .  1,
6  6               | |    .  .  1,
6  7               | |    .  .  1,
6  8               | |    .  .  .  1,
6  9               | |    .  .  .  1,
6  10              | |    .  .  .  .  1,
6  11              |_|    .  .  .  .  .  1;
...
(End)

Links

Crossrefs

Row n has length A138137(n).
Rows sums give A138879.
Cf. A000041, A135010, A138879, A138880, A141285, A182703, A194812, A206437, A211009.

Programs

  • Mathematica
    less[run1_, run2_] := (lg1 = run1 // Length; lg2 = run2 // Length; lg = Max[lg1, lg2]; r1 = If[lg1 == lg, run1, PadRight[run1, lg, 0]]; r2 = If[lg2 == lg, run2, PadRight[run2, lg, 0]]; Order[r1, r2] != -1); row[n_] := Join[Array[1 &, {PartitionsP[n - 1]}], Sort[Reverse /@ Select[IntegerPartitions[n], FreeQ[#, 1] &], less]] // Flatten // Reverse; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 15 2013 *)
Table[Reverse/@Reverse@DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x]!=1]], x_ /; x==0, 2]~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 1, 9}]  // Flatten (* Robert Price, May 11 2020 *)

A141285 Largest part of the n-th partition of j in the list of colexicographically ordered partitions of j, if 1 <= n <= A000041(j).

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 4, 3, 6, 3, 5, 4, 7, 2, 4, 3, 6, 5, 4, 8, 3, 5, 4, 7, 3, 6, 5, 9, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12
Offset: 1

Views

Author

Omar E. Pol, Aug 01 2008

Keywords

Comments

Also largest part of the n-th region of the set of partitions of j, if 1 <= n <= A000041(j). For the definition of "region of the set of partitions of j" see A206437.
Also triangle read by rows: T(j,k) is the largest part of the k-th region in the last section of the set of partitions of j.
For row n >= 2 the rows of triangle are also the branches of a tree which is a projection of a three-dimensional structure of the section model of partitions of A135010, version tree. The branches of even rows give A182730. The branches of odd rows give A182731. Note that each column contains parts of the same size. It appears that the structure of A135010 is a periodic table of integer partitions. See also A210979 and A210980.
Also column 1 of: A193870, A206437, A210941, A210942, A210943. - Omar E. Pol, Sep 01 2013
Also row lengths of A211009. - Omar E. Pol, Feb 06 2014

Examples

			Written as a triangle T(j,k) the sequence begins:
  1;
  2;
  3;
  2, 4;
  3, 5;
  2, 4, 3, 6;
  3, 5, 4, 7;
  2, 4, 3, 6, 5, 4, 8;
  3, 5, 4, 7, 3, 6, 5, 9;
  2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10;
  3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8,  7, 6, 11;
  ...
  ------------------------------------------
  n  A000041                a(n)
  ------------------------------------------
   1 = p(1)                   1
   2 = p(2)                 2 .
   3 = p(3)                   . 3
   4                        2 .
   5 = p(4)               4   .
   6                          . 3
   7 = p(5)                   .   5
   8                        2 .
   9                      4   .
  10                    3     .
  11 = p(6)           6       .
  12                          . 3
  13                          .   5
  14                          .     4
  15 = p(7)                   .       7
  ...
From _Omar E. Pol_, Aug 22 2013: (Start)
Illustration of initial terms (n = 1..11) in three ways: as the largest parts of the partitions of 6 (see A026792), also as the largest parts of the regions of the diagram, also as the diagonal of triangle. By definition of "region" the largest part of the n-th region is also the largest part of the n-th partition (see below):
  --------------------------------------------------------
  .                  Diagram         Triangle in which
  Partitions       of regions       rows are partitions
  of 6           and partitions   and columns are regions
  --------------------------------------------------------
  .                _ _ _ _ _ _
  6                _ _ _      |                         6
  3+3              _ _ _|_    |                       3 3
  4+2              _ _    |   |                     4   2
  2+2+2            _ _|_ _|_  |                   2 2   2
  5+1              _ _ _    | |                 5       1
  3+2+1            _ _ _|_  | |               3 1       1
  4+1+1            _ _    | | |             4   1       1
  2+2+1+1          _ _|_  | | |           2 2   1       1
  3+1+1+1          _ _  | | | |         3   1   1       1
  2+1+1+1+1        _  | | | | |       2 1   1   1       1
  1+1+1+1+1+1       | | | | | |     1 1 1   1   1       1
  ...
The equivalent sequence for compositions is A001511. Explanation: for the positive integer j the diagram of regions of the set of compositions of j has 2^(j-1) regions. The largest part of the n-th region is A001511(n). The number of parts is A006519(n). On the other hand the diagram of regions of the set of partitions of j has A000041(j) regions. The largest part of the n-th region is a(n) = A001511(A228354(n)). The number of parts is A194446(n). Both diagrams have j sections. The diagram for partitions can be interpreted as one of the three views of a three dimensional diagram of compositions in which the rows of partitions are in orthogonal direction to the rest. For the first five sections of the diagrams see below:
  --------------------------------------------------------
  .          Diagram                           Diagram
  .         of regions                        of regions
  .      and compositions                   and partitions
  ---------------------------------------------------------
  .      j = 1 2 3 4 5                     j = 1 2 3 4 5
  ---------------------------------------------------------
   n  A001511                    A228354  a(n)
  ---------------------------------------------------------
   1   1     _| | | | | ............ 1    1    _| | | | |
   2   2     _ _| | | | ............ 2    2    _ _| | | |
   3   1     _|   | | |    ......... 4    3    _ _ _| | |
   4   3     _ _ _| | | ../  ....... 6    2    _ _|   | |
   5   1     _| |   | |    / ....... 8    4    _ _ _ _| |
   6   2     _ _|   | | ../ /   .... 12   3    _ _ _|   |
   7   1     _|     | |    /   /   . 16   5    _ _ _ _ _|
   8   4     _ _ _ _| | ../   /   /
   9   1     _| | |   |      /   /
  10   2     _ _| |   |     /   /
  11   1     _|   |   |    /   /
  12   3     _ _ _|   | ../   /
  13   1     _| |     |      /
  14   2     _ _|     |     /
  15   1     _|       |    /
  16   5     _ _ _ _ _| ../
  ...
Also we can draw an infinite Dyck path in which the n-th odd-indexed line segment has a(n) up-steps and the n-th even-indexed line segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two successive valleys at height 0 is also the partition number A000041(n). See below:
.                                 5
.                                 /\                 3
.                   4            /  \           4    /\
.                   /\          /    \          /\  /
.         3        /  \     3  /      \        /  \/
.    2    /\   2  /    \    /\/        \   2  /
. 1  /\  /  \  /\/      \  /            \  /\/
. /\/  \/    \/          \/              \/
.
.(End)

Links

Crossrefs

Where records occur give A000041, n>=1. Column 1 is A158478. Row j has length A187219(j). Row sums give A138137. Right border gives A000027.
Cf. A000041, A135010, A182730, A182731, A182732, A182733, A182982, A182983, A182703, A193870, A194446, A194447, A194550, A206437, A210979, A210980, A211978, A220517, A225600, A225610.

Programs

  • Mathematica
    Last/@DeleteCases[DeleteCases[Sort@PadRight[Reverse/@IntegerPartitions[13]], x_ /; x == 0, 2], {}] (* updated _Robert Price, May 15 2020 *)

Formula

a(n) = A001511(A228354(n)). - Omar E. Pol, Aug 22 2013

Extensions

Edited by Omar E. Pol, Nov 28 2010
Better definition and edited by Omar E. Pol, Oct 17 2013

A182982 Triangle read by rows: row n lists the parts of the n-th shell of the table A182742.

Original entry on oeis.org

2, 2, 4, 2, 2, 3, 3, 6, 2, 2, 2, 2, 3, 5, 4, 4, 8, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 3, 7, 4, 6, 5, 5, 10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 6, 3, 4, 5, 3, 9, 4, 4, 4, 4, 8, 5, 7, 6, 6, 12, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Omar E. Pol, Jan 26 2011

Keywords

Comments

Apparently this is the main table for even numbers of the shell model of partitions. It appears that the table shows an overlapping of all the heads of last sections of partitions of all even numbers. This is the table 2.0 mentioned in A135010, a geometric version of the table A182742. For odd numbers see A182983. The largest parts of the rows of the diagram give A182732.

Examples

			Triangle begins:
2,
2, 4,
2, 2, 3, 3, 6,
2, 2, 2, 2, 3, 5, 4, 4, 8,
2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 3, 7, 4, 6, 5, 5, 10

Crossrefs

Cf. A135010, A141285, A182732, A182742, A182980, A182983.

A182732 The limit of row A182730(n,.) as n-> infinity.

Original entry on oeis.org

2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12, 5, 4, 8, 7, 6, 11, 6, 5, 10, 9, 8, 7, 14, 4, 7, 6, 5, 10, 5, 9, 8, 7, 13, 4, 8, 7, 6, 12, 6, 11, 10, 9, 8, 16, 3, 6, 5, 9, 4, 8, 7, 6, 12, 7, 6, 11, 5, 10, 9, 8, 15, 6, 5, 10, 9, 8, 7, 14, 8, 7, 13, 6, 12, 11, 10, 9, 18
Offset: 1

Views

Author

Omar E. Pol, Nov 28 2010

Keywords

Comments

Largest part of the n-th partition of the table 2.0 mentioned in A135010. For the table 2.0 see A182982.

Links

Crossrefs

One together with where records occur give A182746.
Cf. A135010, A138121, A141285, A151285, A182730, A182731, A182733, A182982, A182983.

A182733 The limit of row A182731(n,.) as n-> infinity.

Original entry on oeis.org

3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11, 4, 7, 6, 5, 10, 5, 9, 8, 7, 13, 3, 6, 5, 9, 4, 8, 7, 6, 12, 7, 6, 11, 5, 10, 9, 8, 15, 5, 4, 8, 7, 6, 11, 6, 5, 10, 9, 8, 7, 14, 5, 9, 8, 7, 13, 7, 6, 12, 11, 10, 9, 17, 4, 7, 6, 5, 10, 5, 9, 8, 7, 13, 4, 8, 7, 6, 12, 6, 11, 10, 9, 8, 16, 7, 6, 11, 5, 10, 9, 8, 15, 9, 8, 7, 14, 7, 13, 12, 11, 10, 19
Offset: 1

Views

Author

Omar E. Pol, Nov 28 2010

Keywords

Comments

Largest part of the n-th partition of the table 2.1 mentioned in A135010. For the table 2.1 see A182983.

Links

Crossrefs

Zero together with where records occur give A182747.
Cf. A135010, A138121, A141285, A182730, A183731, A182732, A182982, A182983.

A182995 Sum of parts of the n-th subsection of the head of the last section of the set of partitions of any odd integer >= 2n+1.

Original entry on oeis.org

3, 7, 18, 44, 82, 158, 303, 507, 873, 1470, 2354, 3756, 5923, 9065, 13815, 20824, 30853, 45365, 66210, 95415, 136696, 194414, 274057, 384136, 535219, 740559, 1019529, 1396212, 1901533, 2577918, 3479291, 4673711, 6253003, 8332767
Offset: 1

Views

Author

Omar E. Pol, Feb 06 2011

Keywords

Comments

The last section of the set of partitions of 2n+1 contains n subsections.
Also first differences of A182737. - Omar E. Pol, Mar 03 2011

Examples

			a(5)=82 because the 5th subsection of the head of the last section of any odd integer >= 11 looks like this:
(11 . . . . . . . . . . )
( 6 . . . . . 5 . . . . )
( 7 . . . . . . 4 . . . )
( 8 . . . . . . . 3 . . )
( 4 . . . 4 . . . 3 . . )
( 5 . . . . 3 . . 3 . . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
There are 21 parts whose sum is 11+6+5+7+4+8+3+4+4+3+5+3+3+2+2+2+2+2+2+2+2 = 11*6 + 2*8 = 82, so a(5) = 82.

Crossrefs

Cf. A135010, A138880, A182737, A182743, A182983, A182993, A182994.

Formula

a(n) = A138880(2n+1) - A138880(2n-1).

Extensions

a(17) corrected and more terms from Omar E. Pol, Mar 03 2011.
a(12) corrected by Georg Fischer, Aug 31 2020

A182993 Number of parts of the n-th subshell of the head of the last section of the set of partitions of any odd integer >= 2n+1.

Original entry on oeis.org

1, 2, 5, 12, 21, 39, 73, 118, 198, 326, 510, 797, 1234, 1854, 2778, 4122, 6014, 8717, 12550, 17849, 25252, 35486, 49447, 68540, 94480, 129378, 176339, 239165, 322676, 433487, 579907, 772318, 1024691, 1354445, 1783504
Offset: 1

Views

Author

Omar E. Pol, Feb 06 2011

Keywords

Comments

The last section of the set of partitions of 2n+1 contains n subshells.
Also first differences of A182735. - Omar E. Pol, Mar 03 2011

Examples

			a(5)=21 because the 5th subshell of the head of the last section of any odd integer >= 11 looks like this:
(11 . . . . . . . . . . )
( 6 . . . . . 5 . . . . )
( 7 . . . . . . 4 . . . )
( 8 . . . . . . . 3 . . )
( 4 . . . 4 . . . 3 . . )
( 5 . . . . 3 . . 3 . . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
The subshell has 21 parts, so a(5)=21.

Crossrefs

Cf. A135010, A138135, A182735, A182743, A182983, A182992, A182995.

Formula

a(n) = A138135(2n+1) - A138135(2n-1).

Extensions

More terms from Omar E. Pol, Mar 03 2011

A194803 Number of parts that are visible in one of the three views of the shell model of partitions version "Tree" with n shells.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 17, 23, 33, 46, 64, 86, 121, 161, 217, 291, 388, 507, 671, 870, 1131, 1458, 1872, 2383, 3042, 3840, 4841, 6076, 7605, 9460, 11765, 14544, 17950, 22073, 27077, 33092, 40395, 49113, 59611, 72162, 87185, 105035, 126366
Offset: 0

Views

Author

Omar E. Pol, Jan 27 2012

Keywords

Comments

The physical model shows each part represented by an object, for example using a cube or a cuboid. In this case the small version of the model shows each part as a cube of side 1 which is labeled with the size of the part. On the same way the large version of the model shows each part as a cuboid of sides 1 x 1 x L where L is the size of the part. The cuboid is labeled with the level of the part. For the sum of parts see A194804. For more information about the shell model see A135010 and A194805.

Examples

			Illustration of one of the three views with seven shells:
1) Small version:
.
Level
1        A182732 <- 6 3 4 2 1 3 5 4 7 -> A182733
2                     3 2 2 1 2 2 3
3                         2 1 2
4                           1
5      Table 2.0            1            Table 2.1
6                           1
7                           1
.
.  A182742  A182982                   A182743  A182983
.  A182992  A182994                   A182993  A182995
.
2) Large version:
.
.                   . . . . 1 . . . .
.                   . . . 1 2 . . . .
.                   . 1 . . 2 1 . . .
.                   . . 1 2 2 . . 1 .
.                   . . . . 2 2 1 . .
.                   1 2 2 3 2 . . . .
.                           2 3 2 2 1
.
The large version shows the parts labeled with the level of the part where "the level of a part" is its position in the partition. In both versions there are 23 parts that are visible, so a(7) = 23. Also using the formula we have a(7) = 7+8+8 = 23.

Crossrefs

Cf. A006128, A096541, A138135, A135010, A138121, A141285, A182732, A182733, A182742, A182743, A182982, A182983, A182992-A182995, A194804, A194805, A210979.

Formula

a(n) = n + A138135(n-1) + A138135(n), if n >= 2.

A194797 Imbalance of the sum of parts of all partitions of n.

Original entry on oeis.org

0, -2, 1, -7, 3, -21, 7, -49, 23, -97, 57, -195, 117, -359, 256, -624, 498, -1086, 909, -1831, 1634, -2986, 2833, -4847, 4728, -7700, 7798, -12026, 12537, -18633, 19745, -28479, 30723, -42955, 47100, -64284, 71136, -95228, 106402, -139718, 157327, -203495
Offset: 1

Views

Author

Omar E. Pol, Jan 31 2012

Keywords

Comments

Consider the three-dimensional structure of the shell model of partitions, version "tree" (see example). Note that only the parts > 1 produce the imbalance. The 1's are located in the central columns therefore they do not produce the imbalance. Note that every column contains exactly the same parts. For more information see A135010.

Examples

			For n = 6 the illustration of the three views of the shell model with 6 shells shows an imbalance (see below):
------------------------------------------------------
Partitions                Tree             Table 1.0
of 6.                    A194805            A135010
------------------------------------------------------
6                   6                     6 . . . . .
3+3                   3                   3 . . 3 . .
4+2                     4                 4 . . . 2 .
2+2+2                     2               2 . 2 . 2 .
5+1                         1   5         5 . . . . 1
3+2+1                       1 3           3 . . 2 . 1
4+1+1                   4   1             4 . . . 1 1
2+2+1+1                   2 1             2 . 2 . 1 1
3+1+1+1                     1 3           3 . . 1 1 1
2+1+1+1+1                 2 1             2 . 1 1 1 1
1+1+1+1+1+1                 1             1 1 1 1 1 1
------------------------------------------------------
.
.                   6 3 4 2 1 3 5
.     Table 2.0     . . . . 1 . .     Table 2.1
.      A182982      . . . 2 1 . .      A182983
.                   . 3 . . 1 2 .
.                   . . 2 2 1 . .
.                   . . . . 1
------------------------------------------------------
The sum of all parts > 1 on the left hand side is 34 and the sum of all parts > 1 on the right hand side is 13, so a(6) = -34 + 13 = -21. On the other hand for n = 6 the first n terms of A138880 are 0, 2, 3, 8, 10, 24 thus a(6) = 0-2+3-8+10-24 = -21.

Links

Crossrefs

Cf. A000041, A002865, A135010, A138121, A138880, A141285, A182710, A182742, A182743, A182746, A182747, A182982, A182983, A182994, A182995, A194796, A194805.

Programs

  • Maple
    with(combinat):
a:= proc(n) option remember;
      n *(-1)^n *(numbpart(n-1)-numbpart(n)) +a(n-1)
    end: a(0):=0:
seq(a(n), n=1..50); # Alois P. Heinz, Apr 04 2012
  • Mathematica
    a[n_] := Sum[(-1)^(k-1)*k*(PartitionsP[k] - PartitionsP[k-1]), {k, 1, n}]; Array[a, 50] (* Jean-François Alcover, Dec 09 2016 *)

Formula

a(n) = Sum_{k=1..n} (-1)^(k-1)*k*(p(k)-p(k-1)), where p(k) is the number of partitions of k.
a(n) = b(1)-b(2)+b(3)-b(4)+b(5)-b(6)...+-b(n), where b(n) = A138880(n).
a(n) ~ -(-1)^n * Pi * sqrt(2) * exp(Pi*sqrt(2*n/3)) / (48*sqrt(n)). - Vaclav Kotesovec, Oct 09 2018

A194804 Sum of parts that are visible in one of the three views of the shell model of partitions version "tree" with n shells.

Original entry on oeis.org

0, 1, 4, 8, 15, 23, 40, 59, 92, 137, 202, 285, 418, 577, 802, 1106, 1511, 2019, 2724, 3598, 4755, 6226, 8107, 10462, 13523, 17280, 22029, 27953, 35350, 44416, 55763, 69579, 86634, 107459, 132914, 163768, 201475, 246841, 301822, 368033, 447790, 543206
Offset: 0

Views

Author

Omar E. Pol, Jan 27 2012

Keywords

Comments

For the number of parts see A194803. For more information about the shell model see A135010 and A194805.

Examples

			Illustration of one of the three views with seven shells:
.
.        A182732 <- 6 3 4 2 1 3 5 4 7 -> A182733
.                   . . . . 1 . . . .
.                   . . . 2 1 . . . .
.      Table 2.0    . 3 . . 1 2 . . .    Table 2.1
.                   . . 2 2 1 . . 3 .
.                   . . . . 1 2 2 . .
.                           1 . . . .
.  A182742  A182982                   A182743  A182983
.  A182992  A182994                   A182993  A182995
.
The sum of parts that are visible is 1+1+1+1+1+1+1+2+2+2+2+2+2+2+3+3+3+3+4+4+5+6+7 = 59, so a(7) = 59. Using the formula we have a(7) = 7+24+28 = 59.

Crossrefs

Cf. A002865, A066186, A135010, A138121, A138880, A182732, A182733, A182742, A182743, A182982, A182983, A182992-A182995, A194803, A194805.

Formula

a(n) = n + A138880(n-1) + A138880(n), if n >= 2.
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