A295193 -id:A295193 - cp's OEIS frontend

A322787 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic multiset partitions of a multiset with d = A027750(n, k) copies of each integer from 1 to n/d.

1, 2, 2, 3, 3, 5, 7, 5, 7, 7, 11, 23, 21, 11, 15, 15, 22, 79, 66, 22, 30, 162, 30, 42, 274, 192, 42, 56, 56, 77, 1003, 1636, 1338, 565, 77, 101, 101, 135, 3763, 1579, 135, 176, 19977, 10585, 176, 231, 14723, 43686, 4348, 231, 297, 297, 385, 59663, 298416, 82694, 11582, 385

Offset: 1

Gus Wiseman, Dec 26 2018

			Triangle begins:
   1
   2   2
   3   3
   5   7   5
   7   7
  11  23  21  11
  15  15
  22  79  66  22
  30 162  30
  42 274 192  42
Non-isomorphic representatives of the multiset partitions counted under row 6:
{123456}           {112233}           {111222}           {111111}
{1}{23456}         {1}{12233}         {1}{11222}         {1}{11111}
{12}{3456}         {11}{2233}         {11}{1222}         {11}{1111}
{123}{456}         {112}{233}         {111}{222}         {111}{111}
{1}{2}{3456}       {12}{1233}         {112}{122}         {1}{1}{1111}
{1}{23}{456}       {123}{123}         {12}{1122}         {1}{11}{111}
{12}{34}{56}       {1}{1}{2233}       {1}{1}{1222}       {11}{11}{11}
{1}{2}{3}{456}     {1}{12}{233}       {1}{11}{222}       {1}{1}{1}{111}
{1}{2}{34}{56}     {11}{22}{33}       {11}{12}{22}       {1}{1}{11}{11}
{1}{2}{3}{4}{56}   {11}{23}{23}       {1}{12}{122}       {1}{1}{1}{1}{11}
{1}{2}{3}{4}{5}{6} {1}{2}{1233}       {1}{2}{1122}       {1}{1}{1}{1}{1}{1}
                   {12}{13}{23}       {12}{12}{12}
                   {1}{23}{123}       {2}{11}{122}
                   {2}{11}{233}       {1}{1}{1}{222}
                   {1}{1}{2}{233}     {1}{1}{12}{22}
                   {1}{1}{22}{33}     {1}{1}{2}{122}
                   {1}{1}{23}{23}     {1}{2}{11}{22}
                   {1}{2}{12}{33}     {1}{2}{12}{12}
                   {1}{2}{13}{23}     {1}{1}{1}{2}{22}
                   {1}{2}{3}{123}     {1}{1}{2}{2}{12}
                   {1}{1}{2}{2}{33}   {1}{1}{1}{2}{2}{2}
                   {1}{1}{2}{3}{23}
                   {1}{1}{2}{2}{3}{3}

Andrew Howroyd, Table of n, a(n) for n = 1..207 (rows 1..50)

Row sums are A306017. First column is A000041.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306018, A318951, A319190, A319612, A322784, A322785, A322788, A322792.

\\ See A318951 for RowSumMats
row(n)={my(d=divisors(n)); vector(#d, i, RowSumMats(n/d[i], n, d[i]))}
{ for(n=1, 15, print(row(n))) } \\ Andrew Howroyd, Feb 02 2022

Terms a(28) and beyond from Andrew Howroyd, Feb 02 2022

Name edited by Peter Munn, Mar 05 2025

A322789 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic 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, 3, 4, 3, 2, 2, 4, 7, 6, 4, 2, 2, 4, 10, 8, 4, 3, 7, 3, 4, 12, 8, 4, 2, 2, 6, 32, 35, 31, 18, 6, 2, 2, 4, 21, 10, 4, 4, 47, 29, 4, 5, 49, 72, 19, 5, 2, 2, 6, 81, 170, 71, 24, 6, 2, 2, 6, 138, 478, 296, 32, 6, 4, 429, 76, 4, 4, 64, 14, 4

Offset: 1

Gus Wiseman, Dec 26 2018

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

			Triangle begins:
  1
  2  2
  2  2
  3  4  3
  2  2
  4  7  6  4
  2  2
  4 10  8  4
  3  7  3
  4 12  8  4
Non-isomorphic representatives of the multiset partitions counted under row 6:
{123456}           {112233}           {111222}           {111111}
{123}{456}         {112}{233}         {111}{222}         {111}{111}
{12}{34}{56}       {123}{123}         {112}{122}         {11}{11}{11}
{1}{2}{3}{4}{5}{6} {11}{22}{33}       {11}{12}{22}       {1}{1}{1}{1}{1}{1}
                   {11}{23}{23}       {12}{12}{12}
                   {12}{13}{23}       {1}{1}{1}{2}{2}{2}
                   {1}{1}{2}{2}{3}{3}

Andrew Howroyd, Table of n, a(n) for n = 1..338 (rows 1..75)

Row sums are A319056. First column is A000005.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306017, A306018, A319190, A319612, A322784, A322785, A322787, A322792.

Terms a(28) and beyond from Andrew Howroyd, Feb 03 2022

Name edited by Peter Munn, Mar 05 2025

A339847 The number of labeled 6-regular graphs on n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 105, 30016, 11180820, 5188453830, 2977635137862, 2099132870973600, 1803595358964773088, 1872726690127181663775, 2329676580698022197516875, 3443086402825299720403673760, 5997229769947050271535917422040, 12218901113752712984458458475480428

Offset: 0

Atabey Kaygun, Dec 21 2020

nonn

Marni Mishna, Table of n, a(n) for n = 0..195 (terms 0..36 from Andrew Howroyd, terms 37..40 from Atabey Kaygun)
Frédéric Chyzak and Marni Mishna Differential equations satisfied by generating functions of 5-, 6-, and 7-regular labelled graphs: a reduction-based approach, arXiv:2406.04753 [math.CO], 2024.
Atabey Kaygun, Counting Graphs with a Prescribed Degree Sequence.
Atabey Kaygun, Common LISP program that generates the sequence.
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
Marni Mishna, Maple code to generate terms.

Column k=6 of A059441.

Cf. A165627 (unlabeled case), A295193.

\\ Needs GraphsByDegreeSeq from links in A295193.
a(n)={my(M=GraphsByDegreeSeq(n, 6, (p,r)->6-valuation(p,x) <= r)); if(n>=7, vecsum(M[,2]), n==0)} \\ Andrew Howroyd, Dec 26 2020

Terms a(14) and beyond from Andrew Howroyd, Dec 26 2020

A339987 The number of labeled graphs on 2n vertices that share the same degree sequence as any unrooted binary tree on 2n vertices.

Original entry on oeis.org

1, 4, 90, 8400, 1426950, 366153480, 134292027870, 67095690261600, 43893900947947050, 36441011093916429000, 37446160423265535041100, 46669357647008722700474400, 69367722399061403579194432500, 121238024532751529573125745790000, 246171692450596203263023527657431250

Offset: 1

Atabey Kaygun, Dec 25 2020

nonn

An unrooted binary tree is a tree in which all non-leaf vertices have degree 3. With 2n vertices there will be n+1 leaves and n-1 internal vertices.

Andrew Howroyd, Table of n, a(n) for n = 1..100 (terms 1..40 from Atabey Kaygun)
Atabey Kaygun, Counting Graphs with a Prescribed Degree Sequence.
Atabey Kaygun, Common LISP program that generates the sequence.
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Cf. A001147, A001190, A002829.

\\ See Links in A295193 for GraphsByDegreeSeq.
a(n) = {if(n==1, 1, my(d=2*n-4, M=GraphsByDegreeSeq(n-1, 3, (p,r)-> subst(deriv(p),x,1) >= d-6*r), z=(2*n)!/(n-1)!); sum(i=1, matsize(M)[1], my(p=M[i,1], r=(subst(deriv(p), x, 1)-d)/2); M[i,2]*z / (2^polcoef(p,1) * 6^polcoef(p,0) * 2^r * r!)))} \\ Andrew Howroyd, Mar 01 2023

Conjectured recurrence: 32*(1 + n)*(2 + n)*(1 + 2*n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(11589 + 10844*n + 3300*n^2 + 328*n^3)*a(n) - 8*(2 + n)*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(148119 + 232328*n + 129460*n^2 + 30664*n^3 + 2624*n^4)*a(n+1) - 16*(3 + n)*(5 + 2*n)*(7 + 2*n)*(9 + 2*n)*(341634 + 712135*n + 569267*n^2 + 219308*n^3 + 40852*n^4 + 2952*n^5)*a(n+2) + 8*(4 + n)*(7 + 2*n)*(9 + 2*n)*(527520 + 1057879*n + 818282*n^2 + 306380*n^3 + 55672*n^4 + 3936*n^5)*a(n+3) - 2*(5 + n)*(9 + 2*n)*(601452 + 1117119*n + 786236*n^2 + 264028*n^3 + 42472*n^4 + 2624*n^5)*a(n+4) + 3*(4 + n)*(6 + n)*(3717 + 5228*n + 2316*n^2 + 328*n^3)*a(n+5) = 0. - Manuel Kauers and Christoph Koutschan, Mar 01 2023

Conjecture: a(n) ~ 2^(5*n - 1/2) * n^(2*n - 3/2) / (sqrt(Pi) * 3^(n-1) * exp(2*n + 21/16)), based on the recurrence by Manuel Kauers and Christoph Koutschan. - Vaclav Kotesovec, Mar 07 2023

A351264 Number of oriented graphs on n labeled nodes whose underlying graph is regular.

Original entry on oeis.org

1, 1, 3, 9, 125, 1409, 134649, 9775233, 7056026001, 4365949239297, 25236968856946209, 97871747641444333569, 6214784730407041487304385, 252021162589018039813851873281, 105775796091843258520649528395263873, 37978893525907253948045960222247183810561

Offset: 0

Andrew Howroyd, Feb 05 2022

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..24

The unlabeled version is A350913.
Row sums of A351263.
Cf. A295193.

A319877 Numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

Original entry on oeis.org

1, 7, 9, 14, 18, 23, 25, 28, 36, 46, 50, 56, 72, 92, 97, 100, 112, 121, 144, 151, 161, 169, 175, 183, 184, 185, 194, 195, 200, 207, 224, 225, 227, 242, 288, 289, 302, 322, 338, 350, 366, 368, 370, 388, 390, 400, 414, 448, 450, 454, 484, 541, 576, 578, 604, 644

Offset: 1

Gus Wiseman, Dec 17 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of 2-regular multiset multisystems (meaning all vertex-degrees are 2).

			The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
    1: {}
    7: {{1,1}}
    9: {{1},{1}}
   14: {{},{1,1}}
   18: {{},{1},{1}}
   23: {{2,2}}
   25: {{2},{2}}
   28: {{},{},{1,1}}
   36: {{},{},{1},{1}}
   46: {{},{2,2}}
   50: {{},{2},{2}}
   56: {{},{},{},{1,1}}
   72: {{},{},{},{1},{1}}
   92: {{},{},{2,2}}
   97: {{3,3}}
  100: {{},{},{2},{2}}
  112: {{},{},{},{},{1,1}}
  121: {{3},{3}}
  144: {{},{},{},{},{1},{1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  183: {{1},{1,2,2}}
  184: {{},{},{},{2,2}}
  185: {{2},{1,1,2}}
  194: {{},{3,3}}
  195: {{1},{2},{1,2}}
  200: {{},{},{},{2},{2}}

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319878, A319899, A320325, A322526, A322527, A322530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 5, 26, 472, 23342

Offset: 0

Gus Wiseman, Dec 23 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is k-uniform if all edges contain exactly k vertices, and k-regular if all vertices belong to exactly k edges. The span of a hypergraph is the union of its edges.

			The a(3) = 2 hypergraphs:
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
The a(4) = 5 hypergraphs:
  {{1},{2},{3},{4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
The a(5) = 26 hypergraphs:
  {{1},{2},{3},{4},{5}}
  {{1,2},{1,3},{2,4},{3,5},{4,5}}
  {{1,2},{1,3},{2,5},{3,4},{4,5}}
  {{1,2},{1,4},{2,3},{3,5},{4,5}}
  {{1,2},{1,4},{2,5},{3,4},{3,5}}
  {{1,2},{1,5},{2,3},{3,4},{4,5}}
  {{1,2},{1,5},{2,4},{3,4},{3,5}}
  {{1,3},{1,4},{2,3},{2,5},{4,5}}
  {{1,3},{1,4},{2,4},{2,5},{3,5}}
  {{1,3},{1,5},{2,3},{2,4},{4,5}}
  {{1,3},{1,5},{2,4},{2,5},{3,4}}
  {{1,4},{1,5},{2,3},{2,4},{3,5}}
  {{1,4},{1,5},{2,3},{2,5},{3,4}}
  {{1,2,3},{1,2,4},{1,3,5},{2,4,5},{3,4,5}}
  {{1,2,3},{1,2,4},{1,4,5},{2,3,5},{3,4,5}}
  {{1,2,3},{1,2,5},{1,3,4},{2,4,5},{3,4,5}}
  {{1,2,3},{1,2,5},{1,4,5},{2,3,4},{3,4,5}}
  {{1,2,3},{1,3,4},{1,4,5},{2,3,5},{2,4,5}}
  {{1,2,3},{1,3,5},{1,4,5},{2,3,4},{2,4,5}}
  {{1,2,4},{1,2,5},{1,3,4},{2,3,5},{3,4,5}}
  {{1,2,4},{1,2,5},{1,3,5},{2,3,4},{3,4,5}}
  {{1,2,4},{1,3,4},{1,3,5},{2,3,5},{2,4,5}}
  {{1,2,4},{1,3,5},{1,4,5},{2,3,4},{2,3,5}}
  {{1,2,5},{1,3,4},{1,3,5},{2,3,4},{2,4,5}}
  {{1,2,5},{1,3,4},{1,4,5},{2,3,4},{2,3,5}}
  {{1,2,3,4},{1,2,3,5},{1,2,4,5},{1,3,4,5},{2,3,4,5}}

Row sums of A322706.

Cf. A005176, A058891, A059441, A116539, A295193, A299353, A306021, A319056, A319189, A319190, A319612, A321721, A322704.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{k}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,n}],{n,1,6}]

A319878 Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

Original entry on oeis.org

1, 7, 9, 23, 25, 97, 121, 151, 161, 169, 175, 183, 185, 195, 207, 225, 227, 289, 541, 661, 679, 687, 781, 841, 847, 873, 957, 961, 1009, 1089, 1193, 1427, 1563, 1589, 1681, 1819, 1849, 1879, 1895, 2023, 2043, 2167, 2193, 2209, 2231, 2425, 2437, 2585, 2601

Offset: 1

Gus Wiseman, Dec 17 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of 2-regular (all vertex-degrees are 2) multiset partitions (no empty parts).

			The sequence of multiset partitions whose MM-numbers belong to the sequence begins:
    1: {}
    7: {{1,1}}
    9: {{1},{1}}
   23: {{2,2}}
   25: {{2},{2}}
   97: {{3,3}}
  121: {{3},{3}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  183: {{1},{1,2,2}}
  185: {{2},{1,1,2}}
  195: {{1},{2},{1,2}}
  207: {{1},{1},{2,2}}
  225: {{1},{1},{2},{2}}
  227: {{4,4}}
  289: {{4},{4}}
  541: {{1,1,3,3}}
  661: {{5,5}}
  679: {{1,1},{3,3}}
  687: {{1},{1,3,3}}
  781: {{3},{1,1,3}}
  841: {{1,3},{1,3}}
  847: {{1,1},{3},{3}}
  873: {{1},{1},{3,3}}
  957: {{1},{3},{1,3}}
  961: {{5},{5}}

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319877, A319899, A320325, A322526, A322527, A322530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322555 Number of labeled simple graphs on n vertices where all non-isolated vertices have the same degree.

Original entry on oeis.org

1, 1, 2, 5, 18, 69, 390, 2703, 59474, 1548349, 168926258, 12165065351, 7074423247562, 2294426405580191, 4218009215702391954, 3810376434461484994317, 35102248193591661086921250, 156873334244228518638713087133, 4144940994226400702145709978234154

Offset: 0

Gus Wiseman, Dec 15 2018

nonn

Such graphs may be said to have regular support.

			The a(4) = 18 edge sets:
  {}
  {{1,2}}
  {{1,3}}
  {{1,4}}
  {{2,3}}
  {{2,4}}
  {{3,4}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A000569, A005176, A038041, A059441, A295193, A306017, A306019, A319169, A320458.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],SameQ@@Length/@Split[Sort[Join@@#]]&]],{n,6}]

a(n) = 1 + Sum_{k=1..n} binomial(n, k)*(A295193(k) - 1). - Andrew Howroyd, Dec 17 2018

a(8)-a(15) from Andrew Howroyd, Dec 17 2018

a(16)-a(18) from Andrew Howroyd, May 21 2020

A322706 Regular triangle read by rows where T(n,k) is the number of k-regular k-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 12, 12, 1, 0, 1, 70, 330, 70, 1, 0, 1, 465, 11205, 11205, 465, 1, 0, 1, 3507, 505505, 2531200, 505505, 3507, 1, 0

Offset: 1

Gus Wiseman, Dec 23 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is k-uniform if all edges contain exactly k vertices, and k-regular if all vertices belong to exactly k edges. The span of a hypergraph is the union of its edges.

			Triangle begins:
  1
  1       0
  1       1       0
  1       3       1       0
  1      12      12       1       0
  1      70     330      70       1       0
  1     465   11205   11205     465       1       0
  1    3507  505505 2531200  505505    3507       1       0
Row 4 counts the following hypergraphs:
  {{1}{2}{3}{4}}  {{12}{13}{24}{34}}  {{123}{124}{134}{234}}
                  {{12}{14}{23}{34}}
                  {{13}{14}{23}{24}}

Row sums are A322705. Second column is A001205. Third column is A110101.

Cf. A005176, A058891, A059441, A295193, A306021, A319056, A319189, A319190, A319612, A321721, A322704.

Table[Table[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{k}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,n}],{n,1,6}]

cp's OEIS Frontend

A322787 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic multiset partitions of a multiset with d = A027750(n, k) copies of each integer from 1 to n/d.

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A322789 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

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A339847 The number of labeled 6-regular graphs on n nodes.

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A339987 The number of labeled graphs on 2n vertices that share the same degree sequence as any unrooted binary tree on 2n vertices.

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A351264 Number of oriented graphs on n labeled nodes whose underlying graph is regular.

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A319877 Numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

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Mathematica

A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

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A319878 Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

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Mathematica

A322555 Number of labeled simple graphs on n vertices where all non-isolated vertices have the same degree.

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