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.

A001255 Squares of partition numbers.

Original entry on oeis.org

1, 1, 4, 9, 25, 49, 121, 225, 484, 900, 1764, 3136, 5929, 10201, 18225, 30976, 53361, 88209, 148225, 240100, 393129, 627264, 1004004, 1575025, 2480625, 3833764, 5934096, 9060100, 13823524, 20839225, 31404816, 46812964, 69705801, 102880449, 151536100
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000041, A000290, A304990, A200089.
Cf. A133042, A054440.

Programs

Formula

a(n) = A000041(n)^2.
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (48*n^2). - Vaclav Kotesovec, Dec 01 2015
Sum_{n>=1} 1/a(n) = A200089. - Amiram Eldar, May 01 2021
a(n) = A006907(n) + A051748(n) + A051749(n). - R. J. Mathar, Mar 09 2022
a(n) = [(x*y)^n] Product_{k>=1} 1 / ((1 - x^k) * (1 - y^k)). - Ilya Gutkovskiy, Apr 24 2025

Extensions

Extended by Ray Chandler, Nov 14 2005

A366132 Number of unordered pairs of distinct strict integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 6, 10, 15, 28, 45, 66, 105, 153, 231, 351, 496, 703, 1035, 1431, 2016, 2850, 3916, 5356, 7381, 10011, 13530, 18336, 24531, 32640, 43660, 57630, 75855, 100128, 130816, 170820, 222778, 288420, 372816, 481671, 618828, 793170, 1016025, 1295245
Offset: 0

Views

Author

Gus Wiseman, Oct 08 2023

Keywords

Examples

			The a(3) = 1 through a(8) = 15 pairs of strict partitions:
  {3,21}  {4,31}  {5,32}   {6,42}    {7,43}    {8,53}
                  {5,41}   {6,51}    {7,52}    {8,62}
                  {41,32}  {51,42}   {7,61}    {8,71}
                           {6,321}   {52,43}   {62,53}
                           {42,321}  {61,43}   {71,53}
                           {51,321}  {61,52}   {71,62}
                                     {7,421}   {8,431}
                                     {43,421}  {8,521}
                                     {52,421}  {53,431}
                                     {61,421}  {53,521}
                                               {62,431}
                                               {62,521}
                                               {71,431}
                                               {71,521}
                                               {521,431}

Crossrefs

For subsets instead of partitions we have A006516, non-disjoint A003462.
The disjoint case is A108796, non-strict A260669.
For non-strict partitions we have A355389.
The ordered disjoint case is A365662, non-strict A054440.
The ordered version is 2*a(n).
Including equal pairs or twins gives A366317, ordered A304990.
A000041 counts integer partitions, strict A000009.
A002219 and A237258 count partitions of 2n including a partition of n.
A161680 and A000217 count 2-subsets of {1..n}.
Cf. A000124, A001255, A006827, A032302, A064914, A365661, A365663.

Programs

  • Mathematica
    Table[Length[Subsets[Select[IntegerPartitions[n],UnsameQ@@#&],{2}]],{n,0,30}]

Formula

a(n) = binomial(A000009(n),2).

A366317 Number of unordered pairs of strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 10, 15, 21, 36, 55, 78, 120, 171, 253, 378, 528, 741, 1081, 1485, 2080, 2926, 4005, 5460, 7503, 10153, 13695, 18528, 24753, 32896, 43956, 57970, 76245, 100576, 131328, 171405, 223446, 289180, 373680, 482653, 619941, 794430, 1017451, 1296855
Offset: 0

Views

Author

Gus Wiseman, Oct 08 2023

Keywords

Examples

			The a(1) = 1 through a(7) = 15 unordered pairs of strict partitions:
  {1,1}  {2,2}  {3,3}    {4,4}    {5,5}    {6,6}      {7,7}
                {3,21}   {4,31}   {5,32}   {6,42}     {7,43}
                {21,21}  {31,31}  {5,41}   {6,51}     {7,52}
                                  {32,32}  {42,42}    {7,61}
                                  {32,41}  {42,51}    {43,43}
                                  {41,41}  {51,51}    {43,52}
                                           {6,321}    {43,61}
                                           {42,321}   {52,52}
                                           {51,321}   {52,61}
                                           {321,321}  {61,61}
                                                      {7,421}
                                                      {43,421}
                                                      {52,421}
                                                      {61,421}
                                                      {421,421}

Crossrefs

For non-strict partitions we have A086737.
The disjoint case is A108796, non-strict A260669.
The ordered version is A304990, disjoint A032302.
The ordered disjoint case is A365662.
Excluding constant pairs gives A366132.
A000041 counts integer partitions, strict A000009.
A002219 and A237258 count partitions of 2n including a partition of n.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A000712, A007582, A054440, A064914, A260664.

Programs

  • Mathematica
    Table[Length[Select[Tuples[Select[IntegerPartitions[n], UnsameQ@@#&],2],OrderedQ]],{n,0,30}]

Formula

a(n) = A000217(A000009(n)).
Composition of A000009 and A000217.

A304987 G.f.: Sum_{k>=0} A000041(k) * x^k / Sum_{k>=0} A000009(k)^2 * x^k.

Original entry on oeis.org

1, 0, 1, -2, 2, -6, 3, -16, 17, -34, 47, -78, 153, -178, 373, -530, 954, -1410, 2280, -3896, 5908, -9988, 15170, -25908, 40659, -65136, 105967, -169056, 276483, -435624, 712052, -1139814, 1839535, -2955466, 4745201, -7689672, 12303439, -19866340, 31904000
Offset: 0

Views

Author

Vaclav Kotesovec, May 23 2018

Keywords

Crossrefs

Cf. A000009, A000041, A054440, A304877, A304990.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]*x^k, {k, 0, nmax}] / Sum[PartitionsQ[k]^2*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * c * d^n, where d = 1.6096199212376592810929072080593393678131347423108390218672748044914523428584..., c = 0.4455996319406557616008349987776746416976798533740571426884585957313974660...

A304989 G.f.: Sum_{k>=0} A000041(k)^2 * x^k / Sum_{k>=0} A000009(k)^2 * x^k.

Original entry on oeis.org

1, 0, 3, 2, 16, 10, 59, 32, 187, 90, 519, 152, 1439, 164, 3525, -246, 8904, -2500, 21748, -10836, 53918, -36508, 131424, -115266, 328703, -336608, 812615, -957464, 2046225, -2634166, 5152190, -7145682, 13121677, -19039178, 33473773, -50395004, 86035125
Offset: 0

Views

Author

Vaclav Kotesovec, May 23 2018

Keywords

Crossrefs

Cf. A000009, A000041, A001255, A054440, A304877, A304990.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]^2*x^k, {k, 0, nmax}] / Sum[PartitionsQ[k]^2*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * c * d^n, where d = 1.6096199212376592810929072080593393678131347423108390218672748044914523428584..., c = 3.049014588253509415528984781833089943634060493523166258285691300445092167...

A383354 Squares of plane partition numbers.

Original entry on oeis.org

1, 1, 9, 36, 169, 576, 2304, 7396, 25600, 79524, 250000, 737881, 2187441, 6175225, 17363889, 47320641, 127622209, 336135556, 876219201, 2240128900, 5666777284, 14112014436, 34772925625, 84554753089, 203576025636, 484461937089, 1142215875025, 2665572144964, 6166451098756
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 24 2025

Keywords

Links

Crossrefs

Cf. A000219, A001255, A304990.

Programs

  • Mathematica
    nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]^2

Formula

a(n) = [(x*y)^n] Product_{k>=1} 1 / ((1 - x^k) * (1 - y^k))^k.
a(n) = A000219(n)^2.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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