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 1-4 of 4 results.

A172479 Triangle read by rows: T(n,k) = A152827(n)/(A152827(k)* A152827(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 4, 12, 12, 12, 4, 1, 1, 5, 20, 30, 30, 20, 5, 1, 1, 6, 30, 60, 90, 60, 30, 6, 1, 1, 8, 48, 120, 240, 240, 120, 48, 8, 1, 1, 10, 80, 240, 600, 800, 600, 240, 80, 10, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 04 2010

Keywords

Comments

Row sums are 1, 2, 3, 6, 10, 20, 46, 112, 284, 834, 2662, ... .

Examples

			Triangle begins:
  1;
  1,   1;
  1,   1,   1;
  1,   2,   2,   1;
  1,   2,   4,   2,   1;
  1,   3,   6,   6,   3,   1;
  1,   4,  12,  12,  12,   4,   1;
  1,   5,  20,  30,  30,  20,   5,   1;
  1,   6,  30,  60,  90,  60,  30,   6,   1;
  1,   8,  48, 120, 240, 240, 120,  48,   8,   1;
  1,  10,  80, 240, 600, 800, 600, 240,  80,  10,   1;

Links

Crossrefs

Cf. A000009, A152827.

Programs

  • Mathematica
    c[n_] := Product[PartitionsQ[m], {m, 1, n}];
t[n_, m_] := c[n]/(c[m]*c[n - m]);
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

T(n,k) = A152827(n)/(A152827(k)* A152827(n-k)).
T(n,k) = Product_{i=1..n} A000009(i)/(Product_{i=1..k} A000009(i)*Product_{i=1..n-k} A000009(i)).

Extensions

New name and edits by Tom Edgar, Jan 23 2015

A381807 Number of multisets that can be obtained by choosing a constant partition of each m = 0..n and taking the multiset union.

Original entry on oeis.org

1, 1, 2, 4, 12, 24, 92, 184, 704, 2016, 7600, 15200, 80664, 161328, 601696, 2198824, 9868544, 19737088, 102010480, 204020960
Offset: 0

Views

Author

Gus Wiseman, Mar 13 2025

Keywords

Comments

A constant partition is a multiset whose parts are all equal. There are A000005(n) constant partitions of n.

Examples

			The a(1) = 1 through a(4) = 12 multisets:
  {1}  {1,2}    {1,2,3}        {1,2,3,4}
       {1,1,1}  {1,1,1,3}      {1,1,1,3,4}
                {1,1,1,1,2}    {1,2,2,2,3}
                {1,1,1,1,1,1}  {1,1,1,1,2,4}
                               {1,1,1,2,2,3}
                               {1,1,1,1,1,1,4}
                               {1,1,1,1,1,2,3}
                               {1,1,1,1,2,2,2}
                               {1,1,1,1,1,1,1,3}
                               {1,1,1,1,1,1,2,2}
                               {1,1,1,1,1,1,1,1,2}
                               {1,1,1,1,1,1,1,1,1,1}

Crossrefs

The number of possible choices was A066843.
Multiset partitions into constant blocks: A006171, A279784, A295935.
Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
Choosing divisors: A355747, A355733.
Sets of constant multisets with distinct sums: A381635, A381636, A381716.
Strict instead of constant partitions: A381808, A058694, A152827.
A000041 counts integer partitions, strict A000009, constant A000005.
A000688 counts multiset partitions into constant blocks.
A050361 and A381715 count multiset partitions into constant multisets.
A066723 counts partitions coarser than {1..n}, primorial case of A317141.
A265947 counts refinement-ordered pairs of integer partitions.
A321470 counts partitions finer than {1..n}, primorial case of A300383.
Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.

Programs

  • Mathematica
    Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@Range[n]]]],{n,0,10}]

Formula

Primorial case of A381453: a(n) = A381453(A002110(n)).

Extensions

a(16)-a(19) from Christian Sievers, Jun 04 2025

A381808 Number of multisets that can be obtained by choosing a strict integer partition of m for each m = 0..n and taking the multiset union.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 38, 145, 586, 2619, 12096, 58370, 285244, 1436815, 7281062, 37489525, 193417612
Offset: 0

Views

Author

Gus Wiseman, Mar 14 2025

Keywords

Examples

			The a(1) = 1 through a(5) = 12 multisets:
  {1}  {1,2}  {1,2,3}    {1,2,3,4}      {1,2,3,4,5}
              {1,1,2,2}  {1,1,2,2,4}    {1,1,2,2,4,5}
                         {1,1,2,3,3}    {1,1,2,3,3,5}
                         {1,1,1,2,2,3}  {1,1,2,3,4,4}
                                        {1,2,2,3,3,4}
                                        {1,1,1,2,2,3,5}
                                        {1,1,1,2,2,4,4}
                                        {1,1,1,2,3,3,4}
                                        {1,1,2,2,2,3,4}
                                        {1,1,2,2,3,3,3}
                                        {1,1,1,1,2,2,3,4}
                                        {1,1,1,2,2,2,3,3}

Crossrefs

Set systems: A050342, A116539, A296120, A318361.
The number of possible choices was A152827, non-strict A058694.
Set multipartitions with distinct sums: A279785, A381718.
Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
Choosing divisors: A355747, A355733.
Constant instead of strict partitions: A381807, A066843.
A000041 counts integer partitions, strict A000009, constant A000005.
A066723 counts partitions coarser than {1..n}, primorial case of A317141.
A265947 counts refinement-ordered pairs of integer partitions.
A321470 counts partitions finer than {1..n}, primorial case of A300383.
Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.

Programs

  • Mathematica
    Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@Range[n]]]],{n,0,10}]

Extensions

a(12)-a(16) from Christian Sievers, Jun 04 2025

A265097 a(n) = Product_{k=0..n} q(k)^k, where q(k) = partition numbers into distinct parts (A000009).

Original entry on oeis.org

1, 1, 1, 8, 128, 31104, 127401984, 9953280000000, 16717688340480000000, 2243810146471316029440000000, 22438101464713160294400000000000000000, 16671697210628551555613518410547200000000000000000, 2163091659500402360172559530668851200000000000000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 01 2015

Keywords

Crossrefs

Cf. A058694, A152827, A259373.

Programs

  • Mathematica
    Table[Product[PartitionsQ[k]^k, {k,0,n}], {n,0,12}]
Showing 1-4 of 4 results.

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