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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.

User: Olivier Gérard

Olivier Gérard's wiki page.

Olivier Gérard has authored 2676 sequences. Here are the ten most recent ones:

A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

0, 3, 12, 40, 98, 253, 540, 1199, 2415, 4893, 9268, 17864, 32421, 59265, 104632, 184338, 315414, 540155, 901845, 1504173, 2461932, 4013511, 6443170, 10314675, 16281749, 25608450, 39838855, 61716941, 94682665, 144726102
Offset: 1

Views

Author

Olivier Gérard, May 31 2024

Keywords

Comments

The length of each row is given by A000041.
As many sequences start like the nonnegative integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A001477 to have the same row sums for at least 8 terms: A089867, A089868, A089869, A089870, A118760, A123719, A130696, A136602, A254109, A258069, A258070, A258071, A266279, A272813, A273885, A273886, A273887, A273888.

Examples

			Illustration of the first few terms
.
0   | 0
3   | 1,  2
12  | 3,  4,  5
40  | 6,  7,  8,  9,  10
98  | 11, 12, 13, 14, 15, 16, 17
253 | 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28
540 | 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43
.

Crossrefs

Cf. A373300, original version, with positive integers A000027.
Cf. A001477, the nonnegative integers.
Cf. A027480, the sequence of row sums for a regular triangle.

Programs

  • Mathematica
    Module[{s = -1},
 Table[s +=
   PartitionsP[
    n - 1]; (s + PartitionsP[n]) (s + PartitionsP[n] - 1)/2 -
   s (s - 1)/2, {n, 1, 30}]]

A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

1, 5, 15, 45, 105, 264, 555, 1221, 2445, 4935, 9324, 17941, 32522, 59400, 104808, 184569, 315711, 540540, 902335, 1504800, 2462724, 4014513, 6444425, 10316250, 16283707, 25610886, 39841865, 61720659, 94687230, 144731706, 219282679, 330996105, 495901413, 740046425
Offset: 1

Views

Author

Olivier Gérard, May 31 2024

Keywords

Comments

The length of each row is given by A000041.
As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A000027 to have the same row sums for at least 8 terms.
A069782, A088480, A090107, A090108, A090109, A115510, A115511, A130446, A131717, A132086, A153671, A167904, A296879, A296882, A296885, A296888, A296891, A296894, A296897, A296900, A296903, A296906, A303502, A317945, A335280, A361374.

Examples

			Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.
.
    1 |  1
    5 |  2  3
   15 |  4  5  6
   45 |  7  8  9 10 11
  105 | 12 13 14 15 16 17 18
  264 | 19 20 21 22 23 24 25 26 27 28 29
  555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
.
The sequence gives the row sums of this triangle.

Crossrefs

Cf. A000027, seen as a triangle with shape A000041.
Cf. A373301, the same principle, but starting from integer zero instead of 1.
Cf. A006003, row sums of the integers but for the linear triangle.

Programs

  • Mathematica
    Module[{s = 0},
 Table[s +=
   PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 -
   s*(s - 1)/2, {n, 1, 30}]]

A373269 T(n,k) is the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 1
Offset: 1

Views

Author

Olivier Gérard, May 29 2024

Keywords

Comments

The regular array for partitions of n of length k is A373270.
Row sums are A373271.

Examples

			Array begins:
  1,
  1,1,
  1,1,1,
  1,1,1,2,1,
  1,1,1,2,2,2,1,
  1,1,1,2,1,1,2,1,1,2,1,
  1,1,1,2,1,1,2,2,2,2,2,2,2,2,1
  ...
T(10,34) is the first term with value 3. It corresponds to partition 3+2+2+1+1+1 of 10, which has three different multiplicities.

Links

Programs

  • Mathematica
    Flatten@Table[
  Map[Length[Union[Length /@ Split[#]]] &, IntegerPartitions[n]], {n,
   1, 20}]

A373244 T(n,k) = number of integer partitions of n into k parts for which the number of distinct parts is equal to the number of distinct multiplicities.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 3, 2, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 1, 1, 0, 4, 2, 3, 1, 2, 1, 1, 1, 1, 4, 3, 2, 4, 2, 2, 1, 1, 1, 0, 5, 3, 4, 5, 4, 2, 2, 1, 1, 1, 1, 5, 3, 3, 5, 4, 3, 2, 2, 1, 1, 1, 0, 6, 4, 5, 8, 6, 5, 4, 2, 2, 1, 1, 1, 1, 6, 4, 5, 10, 6, 7, 5, 4, 2, 2, 1, 1
Offset: 1

Views

Author

Olivier Gérard, May 29 2024

Keywords

Comments

Row sum is A098859 (Wilf partitions of n).
Counts the zeros in A373241 or A373242.

Examples

			Array begins:
  1,
  1, 1,
  1, 0, 1,
  1, 1, 1, 1,
  1, 0, 2, 1, 1,
  1, 1, 2, 1, 1, 1,
  1, 0, 3, 2, 2, 1, 1,
  1, 1, 3, 2, 2, 2, 1, 1,
  1, 0, 4, 2, 3, 1, 2, 1, 1
  ...

References

  • See references listed in A098859.

Links

Programs

  • Mathematica
    Flatten[Table[
  Plus @@@
   Table[Count[
     Map[Length[Union[#]] == Length[Union[Length /@ Split[#]]] &,
      IntegerPartitions[n, {k}]], True], {k, 1, n}], {n, 1, 20}]]

A373270 Triangle read by rows: T(n,k) is the sum for all integer partitions of n of length k of the number of different multiplicities.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 4, 3, 2, 1, 1, 3, 7, 6, 4, 2, 1, 1, 4, 8, 8, 6, 4, 2, 1, 1, 4, 10, 12, 10, 5, 4, 2, 1, 1, 5, 12, 15, 13, 11, 6, 4, 2, 1, 1, 5, 15, 21, 20, 17, 11, 6, 4, 2, 1, 1, 6, 16, 25, 26, 21, 16, 10, 6, 4, 2, 1, 1, 6, 20, 33, 36, 34, 24, 17, 11, 6, 4, 2, 1, 1, 7, 22, 38, 46, 44, 34, 25, 17, 11, 6, 4, 2, 1, 1, 7, 25, 48, 58, 56, 50, 38, 24, 16, 11, 6, 4, 2, 1
Offset: 1

Views

Author

Olivier Gérard, May 29 2024

Keywords

Examples

			Array begins:
  1,
  1, 1,
  1, 1,  1,
  1, 2,  2,  1,
  1, 2,  4,  2,  1,
  1, 3,  4,  3,  2,  1,
  1, 3,  7,  6,  4,  2,  1,
  1, 4,  8,  8,  6,  4,  2, 1,
  1, 4, 10, 12, 10,  5,  4, 2, 1,
  1, 5, 12, 15, 13, 11,  6, 4, 2, 1,
  1, 5, 15, 21, 20, 17, 11, 6, 4, 2, 1,
  ...
Example of computation:
T(9,3) = 10 because the partitions of 9 into 3 parts are
  7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3,
the number of different multiplicities are
  2, 1, 1, 2, 2, 1, 1,
and the sum of these multiplicities is 10.

Links

Crossrefs

Cf. A373242, A373269, A373271.

Programs

  • Mathematica
    Flatten@Table[
  Plus @@@
   Table[Map[Length[Union[Length /@ Split[#]]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]

A373242 T(n,k) is the sum for all integer partitions of n of length k of the difference between the number of different parts and the number of different multiplicities.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 1, 0, 0, 0, 3, 2, 1, 0, 0, 0, 0, 3, 4, 3, 1, 0, 0, 0, 0, 4, 6, 4, 2, 2, 0, 0, 0, 0, 4, 8, 8, 5, 1, 1, 0, 0, 0, 0, 5, 10, 10, 7, 2, 1, 1, 0, 0, 0, 0, 5, 14, 16, 12, 8, 3, 2, 1, 0, 0, 0, 0, 6, 16, 20, 17, 8, 6, 2, 1, 1, 0, 0, 0, 0, 6, 20, 29, 25, 16, 10, 5, 2, 1, 1, 0, 0, 0, 0, 7, 24, 35, 36, 27, 14, 7, 6, 3, 1, 1, 0, 0, 0
Offset: 1

Views

Author

Olivier Gérard, May 29 2024

Keywords

Comments

The corresponding irregular triangle (one entry for each partition of n) is A373241.
The sum of each row is A373243.
The corresponding triangle for sum of number of different parts is A092905.
The corresponding triangle for sum of number of different multiplicities is A373271.

Examples

			Array begins:
  0
  0,0
  0,1,0
  0,1,0,0
  0,2,0,0,0
  0,2,2,1,0,0
  0,3,2,1,0,0,0
  0,3,4,3,1,0,0,0
  0,4,6,4,2,2,0,0,0
  0,4,8,8,5,1,1,0,0,0
  ...
Example of computation:
T(9,3) = 6 because the partitions of 9 into 3 parts are
  7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3,
the numbers of different parts are
  2, 3, 3, 2, 2, 3, 1,
the numbers of different multiplicities are
  2, 1, 1, 2, 2, 1, 1,
the differences between them are
  0, 2, 2, 0, 0, 2, 0,
and the sum of these differences is 6.

Links

Programs

  • Mathematica
    Flatten[Table[
  Plus @@@
   Table[Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]]

A373243 a(n) = sum for all integer partitions of n of the difference between number of different parts and number of different multiplicities.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 6, 11, 18, 27, 36, 61, 77, 115, 161, 223, 291, 416, 531, 729, 951, 1256, 1605, 2132, 2694, 3491, 4423, 5659, 7079, 9027, 11201, 14102, 17484, 21789, 26822, 33309, 40734, 50160, 61195, 74893, 90846, 110722, 133697, 162026, 195104, 235244
Offset: 1

Views

Author

Olivier Gérard, May 29 2024

Keywords

Comments

Sum of the rows of A373241 or A373242.

Examples

			From the eighth row of A373241: a(8)=11
  0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0
or the tenth row of A373242: a(10)=27
  0, 4, 8, 8, 5, 1, 1, 0, 0, 0

Crossrefs

Cf. A373241, A373242.

Programs

  • Mathematica
    Table[Plus @@
  Table[Plus @@
    Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 40}]

A373271 a(n) = sum for all integer partitions of n of the number of distinct multiplicities in each partition.

Original entry on oeis.org

1, 2, 3, 6, 10, 14, 24, 34, 49, 70, 103, 134, 195, 258, 347, 461, 624, 796, 1066, 1358, 1763, 2250, 2903, 3631, 4644, 5805, 7309, 9083, 11381, 13998, 17428, 21369, 26336, 32174, 39451, 47847, 58399, 70610, 85590, 103077, 124462, 149169, 179368, 214300, 256397
Offset: 1

Views

Author

Olivier Gérard, May 29 2024

Keywords

Comments

Sum of the rows of A373269.
Sum of the rows of A373270.
The multiplicity of a part in an integer partition (or composition) is the number of times it appears in the partition, seen as a list.
The multiplicity of 3 in the partition 12 = 5+3+3+1 is 2.
For this sequence, only distinct multiplicities appearing for parts of the partition are counted, only once for a given partition.
If all multiplicities of all parts of all integer partitions of n are counted, one gets A000070 (1, 2, 4, 7, 12, 19, 30, 45, 67, 97, ...).
If all distinct multiplicities of all parts of all integer partitions are summed, one gets A373273 (1, 3, 5, 11, 18, 29, 48, 74, 107, 161, ...).
If all multiplicities of all parts of all integer partitions of n are summed, one gets A006128 (1, 3, 6, 12, 20, 35, 54, 86, 128, 192, ...).

Examples

			Example for n=20:
the partition 20=4+3+3+3+3+2+1+1
has multiplicities 1, 4, 1, 2
for the parts 4,3,2,1 listed in descending order.
It has 3 different multiplicities (1, 2 and 4) and contributes 3 to a(20) = 1358.

Links

Crossrefs

Cf. A373243, A098859.

Programs

  • Mathematica
    Table[Plus @@
  Table[Plus @@
    Map[Length[Union[Length /@ Split[#]]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 40}]

A373273 a(n) = sum of all distinct multiplicities in every integer partition of n.

Original entry on oeis.org

1, 3, 5, 11, 18, 29, 48, 74, 107, 161, 237, 324, 471, 648, 876, 1199, 1632, 2141, 2881, 3754, 4910, 6389, 8301, 10570, 13603, 17280, 21915, 27614, 34842, 43416, 54415, 67509, 83782, 103459, 127685, 156457, 192155, 234540, 286112, 347571, 422245, 510311, 617216
Offset: 1

Views

Author

Olivier Gérard, May 29 2024

Keywords

Comments

Sum of the rows of triangle A373272.

Crossrefs

Cf. A006128, A373272.

Programs

  • Mathematica
    Table[Plus @@
  Map[Plus @@ Union[Length /@ Split[#]] &, IntegerPartitions[n]], {n,
  1, 40}]

A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 0, 0, 2, 2, 1, 1, 1, 0, 0, 1, 0, 3, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Olivier Gérard, May 29 2024

Keywords

Comments

This difference is always nonnegative.
The number of zero values in each row is A098859.
The number of ones in each row is A325244.
The number of positive entries in each row is A336866.
The corresponding regular triangle for partitions of n of length k is A373242.
The sum of each row is A373243.

Examples

			The array begins
  0
  0,0
  0,1,0
  0,1,0,0,0
  0,1,1,0,0,0,0
  0,1,1,0,0,2,0,0,1,0,0
  0,1,1,0,1,2,0,0,0,1,0,0,0,0,0
  0,1,1,0,1,2,0,0,2,0,1,0,0,1,1,1,0,0,0,0,0,0
  0,1,1,0,1,2,0,1,2,0,1,0,0,2,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0
  ...

Links

Crossrefs

Cf. A373269 a triangle of the same shape and order for number of multiplicities.
Cf. A080577, A098859, A325244, A336866, A373242, A373243.

Programs

  • Mathematica
    Flatten @ Table[
  Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
   IntegerPartitions[n]], {n, 1, 20}]

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