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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A322635 Number of regular graphs with loops on n labeled vertices.

Original entry on oeis.org

2, 4, 4, 24, 78, 1908, 23368, 1961200, 75942758, 25703384940, 4184912454930, 4462909435830552, 2245354417775573206, 10567193418810168583576, 24001585002447984453495392, 348615956932626441906675011568, 2412972383955442904868321667433106, 162906453913051798826796439651249753404
Offset: 1

Views

Author

Gus Wiseman, Dec 21 2018

Keywords

Comments

A graph is regular if all vertices have the same degree. A loop adds 2 to the degree of its vertex.

Links

Crossrefs

Row sums of A333158.
Cf. A000666, A054921, A059441, A295193, A299353, A306017, A306021, A319189, A319190, A319612, A322659, A322661.

Programs

  • Mathematica
    Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Select[Tuples[Range[n],2],OrderedQ]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,2n}],{n,6}]
  • PARI
    for(n=1, 10, print1(A322635(n), ", ")) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

Extensions

a(11)-a(18) from Andrew Howroyd, Aug 28 2019

A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 7, 1, 1, 25, 37, 5, 1, 1, 75, 207, 85, 21, 1, 1, 231, 1347, 525, 591, 7, 1, 1, 763, 10125, 21385, 23551, 3535, 113, 1, 1, 2619, 86173, 180201, 1216701, 31647, 30997, 9, 1, 1, 9495, 819133, 12066705, 77636583, 66620631, 11485825, 286929, 955, 1
Offset: 1

Views

Author

Gus Wiseman, Dec 17 2018

Keywords

Examples

			Triangle begins:
  1
  1       1
  1       3       1
  1       9       7       1
  1      25      37       5       1
  1      75     207      85      21       1
  1     231    1347     525     591       7       1
  1     763   10125   21385   23551    3535     113       1
  1    2619   86173  180201 1216701   31647   30997       9       1

Links

Crossrefs

Row sums are A322555.
Cf. A000569, A005176, A058891, A059441, A295193, A301481, A306017, A306019, A306021, A319169, A319190, A319612.

Programs

  • Mathematica
    Table[If[k==0,1,Sum[Binomial[n,sup]*SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[sup],{2}]}],Sequence@@Table[{x[i],0,k},{i,sup}]],{sup,n}]],{n,8},{k,0,n-1}]

Formula

T(n,k) = Sum_{i=1..n} binomial(n,i)*A059441(i,k) for k > 0. - Andrew Howroyd, Dec 26 2020

A322698 Number of regular graphs with half-edges on n labeled vertices.

Original entry on oeis.org

1, 2, 4, 10, 40, 278, 3554, 84590, 3776280, 317806466, 50710452574, 15414839551538, 8964708979273634, 10008446308186072290, 21518891146915893435358, 89320970210116481106835986, 717558285660687970023516336792, 11176382741327158622885664697124082, 338202509574712032788035618665293979610
Offset: 0

Views

Author

Gus Wiseman, Dec 23 2018

Keywords

Comments

A graph is regular if all vertices have the same degree. A half-edge is like a loop except it only adds 1 to the degree of its vertex.

Examples

			The a(3) = 10 edge sets:
  {}
  {{1},{2,3}}
  {{3},{1,2}}
  {{2},{1,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
  {{1},{3},{1,2},{2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}

Links

Crossrefs

Row sums of A333157.
Cf. A058891, A059441, A116539, A283877, A295193, A319189, A319190, A319612, A319729.

Programs

  • Mathematica
    Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Union/@Select[Tuples[Range[n],2],OrderedQ]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,n-1}],{n,1,6}]
  • PARI
    for(n=1, 10, print1(A322698(n), ", ")) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

Extensions

a(10)-a(18) from Andrew Howroyd, Aug 28 2019

A322784 Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

Original entry on oeis.org

1, 1, 4, 8, 29, 59, 311, 892, 4983, 21863, 126813, 678626, 4446565, 27644538, 195561593, 1384705697, 10613378402, 82864870101, 686673571479, 5832742205547, 51897707277698, 474889512098459, 4514467567213008, 44152005855085601, 446355422070799305, 4638590359349994120
Offset: 0

Views

Author

Gus Wiseman, Dec 26 2018

Keywords

Comments

A multiset is uniform if all multiplicities are equal.
Also the number of factorizations into factors > 1 of primorial powers k in A100778 with sum of prime indices A056239(k) equal to n.
a(n) is the number of nonequivalent nonnegative integer matrices without zero rows or columns with equal column sums and total sum n up to permutation of rows. - Andrew Howroyd, Jan 11 2020

Examples

			The a(1) = 1 through a(4) = 29 multiset partitions:
  {{1}}   {{1,1}}     {{1,1,1}}       {{1,1,1,1}}
          {{1,2}}     {{1,2,3}}       {{1,1,2,2}}
         {{1},{1}}   {{1},{1,1}}      {{1,2,3,4}}
         {{1},{2}}   {{1},{2,3}}     {{1},{1,1,1}}
                     {{2},{1,3}}     {{1,1},{1,1}}
                     {{3},{1,2}}     {{1},{1,2,2}}
                    {{1},{1},{1}}    {{1,1},{2,2}}
                    {{1},{2},{3}}    {{1,2},{1,2}}
                                     {{1},{2,3,4}}
                                     {{1,2},{3,4}}
                                     {{1,3},{2,4}}
                                     {{1,4},{2,3}}
                                     {{2},{1,1,2}}
                                     {{2},{1,3,4}}
                                     {{3},{1,2,4}}
                                     {{4},{1,2,3}}
                                    {{1},{1},{1,1}}
                                    {{1},{1},{2,2}}
                                    {{1},{2},{1,2}}
                                    {{1},{2},{3,4}}
                                    {{1},{3},{2,4}}
                                    {{1},{4},{2,3}}
                                    {{2},{2},{1,1}}
                                    {{2},{3},{1,4}}
                                    {{2},{4},{1,3}}
                                    {{3},{4},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{3},{4}}

Links

Crossrefs

Row sums of A322786.
Cf. A001055, A005176, A056239, A072774, A100778, A219727, A295193, A306017 (unlabeled version), A319190, A319612, A322785, A322786, A322792.

Programs

  • Mathematica
    u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
Table[Sum[u[Array[Prime,d,1,Times]^(n/d),Array[Prime,d,1,Times]^(n/d)],{d,Divisors[n]}],{n,12}]

Formula

a(n) = Sum_{d|n} A001055(A002110(n/d)^d).
a(n) = Sum_{d|n} A219727(n/d, d). - Andrew Howroyd, Jan 11 2020

Extensions

a(14)-a(15) from Alois P. Heinz, Jan 16 2019
Terms a(16) and beyond from Andrew Howroyd, Jan 11 2020

A322528 Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 5, 4, 7, 2, 7, 4, 7, 7, 9, 3, 10, 5, 12, 9, 8, 6, 14, 10, 12, 10, 14, 11, 20, 13, 18, 13, 16, 16, 25, 16, 19, 20, 26, 18, 30, 19, 27, 26, 27, 22, 38, 30, 37, 28, 38, 32, 43, 37, 46, 40, 47, 40, 66, 49, 58, 56, 64, 56, 73, 58, 76, 70, 85
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2018

Keywords

Examples

			The a(1) = 1 through a(8) = 5 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (52)       (44)
                    (1111)  (11111)  (222)     (1111111)  (53)
                                     (111111)             (2222)
                                                          (11111111)

Crossrefs

Cf. A002865, A003963, A005117, A064573 , A072774, A295193, A302505, A306017, A319169, A320322, A322526, A322527, A322529, A322530, A322531.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SameQ@@Last/@FactorInteger[Times@@#]]&]],{n,30}]

Extensions

More terms from Alois P. Heinz, Dec 14 2018

A322659 Number of connected regular simple graphs on n labeled vertices.

Original entry on oeis.org

1, 1, 1, 4, 13, 146, 826, 44808, 1074557, 155741296, 10381741786, 6939251270348, 2203360264480750, 4186526735251514044, 3747344007864300197810, 35041787059621536192399824, 156277111373298355107598128061, 4142122641757597729416733678931968
Offset: 1

Views

Author

Gus Wiseman, Dec 22 2018

Keywords

Comments

A graph is regular if all vertices have the same degree.

Links

Crossrefs

Row sums of A324163.
Cf. A054921, A059441, A062740, A295193, A299353, A305078, A319190, A319612, A319729, A322635.

Programs

  • Mathematica
    csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[If[n==1,1,Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],SameQ@@Length/@Split[Sort[Join@@#]],Length[csm[#]]==1]&]]],{n,6}]

Extensions

a(8)-a(15) from Andrew Howroyd, Dec 23 2018
a(16)-a(18) from Andrew Howroyd, Sep 02 2019

A322788 Irregular triangle read by rows where T(n,k) is the number of uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 2, 2, 5, 4, 3, 2, 2, 27, 11, 6, 4, 2, 2, 142, 29, 8, 4, 282, 12, 3, 1073, 101, 8, 4, 2, 2, 32034, 1581, 234, 75, 20, 6, 2, 2, 136853, 2660, 10, 4, 1527528, 1985, 91, 4, 4661087, 64596, 648, 20, 5, 2, 2, 227932993, 1280333, 41945, 231, 28, 6
Offset: 1

Views

Author

Gus Wiseman, Dec 26 2018

Keywords

Comments

A multiset partition is uniform if all parts have the same size.

Examples

			Triangle begins:
     1
     2    2
     2    2
     5    4    3
     2    2
    27   11    6    4
     2    2
   142   29    8    4
   282   12    3
  1073  101    8    4
The multiset partitions counted under row 6:
  {123456}          {112233}          {111222}          {111111}
  {123}{456}        {112}{233}        {111}{222}        {111}{111}
  {124}{356}        {113}{223}        {112}{122}        {11}{11}{11}
  {125}{346}        {122}{133}        {11}{12}{22}      {1}{1}{1}{1}{1}{1}
  {126}{345}        {123}{123}        {12}{12}{12}
  {134}{256}        {11}{22}{33}      {1}{1}{1}{2}{2}{2}
  {135}{246}        {11}{23}{23}
  {136}{245}        {12}{12}{33}
  {145}{236}        {12}{13}{23}
  {146}{235}        {13}{13}{22}
  {156}{234}        {1}{1}{2}{2}{3}{3}
  {12}{34}{56}
  {12}{35}{46}
  {12}{36}{45}
  {13}{24}{56}
  {13}{25}{46}
  {13}{26}{45}
  {14}{23}{56}
  {14}{25}{36}
  {14}{26}{35}
  {15}{23}{46}
  {15}{24}{36}
  {15}{26}{34}
  {16}{23}{45}
  {16}{24}{35}
  {16}{25}{34}
  {1}{2}{3}{4}{5}{6}

Links

Crossrefs

Row sums are A322785. First column is A038041.
Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306017, A319190, A319612, A322784, A322785, A322786, A322789, A322792.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[mps[Join@@Table[Range[n/d],{d}]],SameQ@@Length/@#&]],{n,10},{d,Divisors[n]}]

Formula

T(n,k) = A322794(A002110(n/d)^d), where d = A027750(n,k).

Extensions

More terms from Alois P. Heinz, Jan 30 2019
Terms a(38) and beyond from Andrew Howroyd, Feb 03 2022
Edited by Peter Munn, Mar 05 2025

A351287 Number of symmetric 0-1 matrices with zero main diagonal and nondecreasing number of ones in the rows.

Original entry on oeis.org

1, 2, 4, 16, 84, 936, 16758, 602544, 37693734, 4588585904, 1016082688298, 436137488655846, 348748058993750616, 538461898813943437676
Offset: 1

Views

Author

Max Alekseyev, Feb 06 2022

Keywords

Comments

Also, number of graphs with vertices labeled 1, 2, ..., n such that their degrees are nondecreasing.

Crossrefs

Cf. A016121, A351157, A351288.

Programs

  • PARI
    \\ See link in A295193 for GraphsByDegreeSeq.
a(n)={my(M=GraphsByDegreeSeq(n,n,(p,r)->1)); sum(i=1, matsize(M)[1], my(u=Vec(M[i,1])); prod(j=1, #u, u[j]!)*M[i,2]/n!)} \\ Andrew Howroyd, Feb 06 2022
  • Sage
    def a351287(n): return sum(prod(factorial(e) for e in Partition((d+1 for d in G.degree_sequence())).to_exp()) // G.automorphism_group(return_group=False, order=True) for G in graphs(n))

Extensions

a(11)-a(14) from Andrew Howroyd, Feb 06 2022

A322704 Number of regular hypergraphs on n labeled vertices with no singletons.

Original entry on oeis.org

1, 1, 2, 4, 80, 209944
Offset: 0

Views

Author

Gus Wiseman, Dec 23 2018

Keywords

Comments

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is regular if all vertices have the same degree.

Examples

			The a(3) = 4 edge-sets:
  {}
  {{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Cf. A058891, A059441, A295193, A306021, A319189, A319190, A319612, A319729, A322698.

Programs

  • Mathematica
    Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2,n}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,2^n-n-1}],{n,1,5}]

A322786 Irregular triangle read by rows where T(n,k) is the number of multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 5, 3, 15, 9, 5, 52, 7, 203, 66, 31, 11, 877, 15, 4140, 712, 109, 22, 21147, 686, 30, 115975, 10457, 339, 42, 678570, 56, 4213597, 198091, 27036, 6721, 1043, 77, 27644437, 101, 190899322, 4659138, 2998, 135, 1382958545, 1688360, 58616, 176
Offset: 1

Views

Author

Gus Wiseman, Dec 26 2018

Keywords

Examples

			Triangle begins:
        1
        2       2
        5       3
       15       9       5
       52       7
      203      66      31      11
      877      15
     4140     712     109      22
    21147     686      30
   115975   10457     339      42
   678570      56
  4213597  198091   27036    6721    1043      77
For example, row 4 counts the following multiset partitions.
  {{1,2,3,4}}        {{1,1,2,2}}        {{1,1,1,1}}
  {{1},{2,3,4}}      {{1},{1,2,2}}      {{1},{1,1,1}}
  {{1,2},{3,4}}      {{1,1},{2,2}}      {{1,1},{1,1}}
  {{1,3},{2,4}}      {{1,2},{1,2}}      {{1},{1},{1,1}}
  {{1,4},{2,3}}      {{2},{1,1,2}}      {{1},{1},{1},{1}}
  {{2},{1,3,4}}      {{1},{1},{2,2}}
  {{3},{1,2,4}}      {{1},{2},{1,2}}
  {{4},{1,2,3}}      {{2},{2},{1,1}}
  {{1},{2},{3,4}}    {{1},{1},{2},{2}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{4},{1,2}}
  {{1},{2},{3},{4}}

Links

Crossrefs

Row sums are A322784. First column is A000110.
Cf. A001055, A005176, A027750, A056239, A072774, A100778, A219727, A295193, A306017, A319190, A319612, A322784, A322785, A322787, A322788, A322792.

Programs

  • Mathematica
    u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
Table[Table[u[Array[Prime,n/d,1,Times]^d,Array[Prime,n/d,1,Times]^d],{d,Divisors[n]}],{n,10}]
  • PARI
    \\ needs T(n,k) from A219727.
Row(n)={[T(d,n/d) | d<-divisors(n)]}
{ for(n=1, 12, print(Row(n))) } \\ Andrew Howroyd, Jan 11 2020

Formula

T(n,k) = A001055(A002110(n/d)^d), where d = A027750(n,k).
T(n,k) = A219727(d, n/d), where d = A027750(n, k). - Andrew Howroyd, Jan 11 2020

Extensions

Edited by Peter Munn, Mar 05 2025
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