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 1-5 of 5 results.

A319190 Number of regular hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 3, 19, 879, 5280907, 1069418570520767
Offset: 0

Views

Author

Gus Wiseman, Dec 17 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. The span of a hypergraph is the union of its edges.

Examples

			The a(3) = 19 regular hypergraphs:
                 {{1,2,3}}
                {{1},{2,3}}
                {{2},{1,3}}
                {{3},{1,2}}
               {{1},{2},{3}}
            {{1},{2,3},{1,2,3}}
            {{2},{1,3},{1,2,3}}
            {{3},{1,2},{1,2,3}}
            {{1,2},{1,3},{2,3}}
           {{1},{2},{3},{1,2,3}}
           {{1},{2},{1,3},{2,3}}
           {{1},{3},{1,2},{2,3}}
           {{2},{3},{1,2},{1,3}}
        {{1,2},{1,3},{2,3},{1,2,3}}
       {{1},{2},{1,3},{2,3},{1,2,3}}
       {{1},{3},{1,2},{2,3},{1,2,3}}
       {{2},{3},{1,2},{1,3},{1,2,3}}
      {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Column sums of A188445.
Cf. A002829, A005176, A049311, A058891, A110100, A110101, A116539, A283877, A295193, A306017, A319189.

Programs

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

Extensions

a(6) from Andrew Howroyd, Mar 12 2020

A319189 Number of uniform regular hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 2, 3, 10, 29, 3780, 5012107
Offset: 0

Views

Author

Gus Wiseman, Dec 17 2018

Keywords

Comments

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is uniform if all edges have the same size, and regular if all vertices have the same degree. The span of a hypergraph is the union of its edges.
Also the number of 0-1 matrices with n columns, all distinct rows, no zero columns, equal row-sums, and equal column-sums, up to a permutation of the rows.

Examples

			The a(4) = 10 edge-sets:
               {{1,2,3,4}}
              {{1,2},{3,4}}
              {{1,3},{2,4}}
              {{1,4},{2,3}}
            {{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}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}
Inequivalent representatives of the a(4) = 10 matrices:
  [1 1 1 1]
.
  [1 1 0 0] [1 0 1 0] [1 0 0 1]
  [0 0 1 1] [0 1 0 1] [0 1 1 0]
.
  [1 0 0 0] [1 1 0 0] [1 1 0 0] [1 0 1 0] [1 1 1 0]
  [0 1 0 0] [1 0 1 0] [1 0 0 1] [1 0 0 1] [1 1 0 1]
  [0 0 1 0] [0 1 0 1] [0 1 1 0] [0 1 1 0] [1 0 1 1]
  [0 0 0 1] [0 0 1 1] [0 0 1 1] [0 1 0 1] [0 1 1 1]
.
  [1 1 0 0]
  [1 0 1 0]
  [1 0 0 1]
  [0 1 1 0]
  [0 1 0 1]
  [0 0 1 1]

Crossrefs

Uniform hypergraphs are counted by A306021. Unlabeled uniform regular multiset partitions are counted by A319056. Regular graphs are A295193. Uniform clutters are A299353.
Cf. A002829, A005176, A007016, A049311, A058891, A101370, A110100, A110101, A321717, A321720.

Programs

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

Extensions

a(7) from Jinyuan Wang, Jun 20 2020

A319612 Number of regular simple graphs spanning n vertices.

Original entry on oeis.org

1, 0, 1, 1, 7, 13, 171, 931, 45935, 1084413, 155862511, 10382960971, 6939278572095, 2203360500122299, 4186526756621772343, 3747344008241368443819, 35041787059691023579970847, 156277111373303386104606663421, 4142122641757598618318165240180095
Offset: 0

Views

Author

Gus Wiseman, Dec 17 2018

Keywords

Comments

A graph is regular if all vertices have the same degree. The span of a graph is the union of its edges.

Examples

			The a(4) = 7 edge-sets:
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{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}}

Crossrefs

Cf. A002829, A005176, A110100, A116539, A295193, A306017, A319189, A319190.

Formula

a(n) = A295193(n) - 1.

Extensions

a(16)-a(18) from Andrew Howroyd, Sep 02 2019

A110101 a(n) is the number of 3-regular 3-hypergraphs on n labeled vertices. (In a 3-hypergraph, each hyper-edge is a proper 3-set; 3-regular implies that each vertex is in exactly 3 hyperedges.)

Original entry on oeis.org

1, 0, 0, 0, 1, 12, 330, 11205, 505505, 28787052, 2024844444, 172592502570, 17545270969545, 2098273032696720, 291739927315433454, 46676360010342811203
Offset: 0

Views

Author

Marni Mishna, Jul 11 2005

Keywords

Comments

P-recursive

Examples

			The 3-regular 3-hypergraphs on 4 vertices: {1,2,3}, {2,3,4},{3,4,1},{4,1,2}.

Links

Crossrefs

Cf. A025035, A110100, A002829.

Formula

Differential equation satisfied by exponential generating function: {F(0) = 1, 36*t^2*(t + 1)*(t^2 - 2)^2*(3*t^2 + 2*t - 2)^2*(d^2/dt^2)F(t) - 12*(t + 1)*(3*t^2 + 2*t - 2)*(3*t^10 + 2*t^9 - 8*t^8 - 40*t^7 - 56*t^6 + 4*t^5 - 48*t^4 - 96*t^3 + 80*t^2 + 80*t - 32)*(d/dt)F(t) + t^3*(t + 1)*(3*t^2 + 2*t - 2)*(3*t^9 + 2*t^8 - 2*t^7 - 108*t^6 - 144*t^5 + 32*t^4 - 24*t^3 + 16*t^2 + 112*t - 64)*F(t)}.
Linear recurrence for a(n): initial values: a(2) = 0, a(3) = 0, a(0) = 1, a(1) = 0, a(4) = 1, a(5) = 12, a(6) = 330, a(7) = 11205, a(8) = 505505, a(9) = 28787052, a(10) = 2024844444, a(11) = 172592502570, a(12) = 17545270969545;
then (1971620508*n^4 + 4242044664*n^3 + 3*n^12 + 4459328640*n + 1437004800 +
167310*n^9 + 5794678656*n^2 + 20779902*n^7 + 234*n^11 + 8151*n^10 + 2248389*n^8
+ 618210450*n^5 + 134970693*n^6)*a(n) + (154*n^10 + 77519860*n^5 + 334620440*n^4
+ 958003200 + 5280*n^9 + 106260*n^8 + 1392666*n^7 + 12460602*n^6 + 979793232*n^3
+ 1848236544*n^2 + 2014882560*n + 2*n^11)*a(n + 1) + ( - 96300*n^7 - 1200066*n^6
- 540148032*n^2 - 767940480*n - 4980*n^8 - 57398920*n^4 - 219822600*n^3
- 479001600 - 10060470*n^5 - 2*n^10 - 150*n^9)*a(n + 2) + ( - 97416*n^8
- 17244057600 - 24771847680*n - 2808*n^9 - 36*n^10 - 1978992*n^7 - 26064612*n^6
- 232501752*n^5 - 1422206064*n^4 - 5889271968*n^3 - 15795689472*n^2)*a(n
+ 3) + ( - 5364230400*n - 4790016000 - 24*n^9 - 1872*n^8 - 64368*n^7 - 1280160*n^6
- 16223256*n^5 - 135808848*n^4 - 750702432*n^3 - 2641118400*n^2)*a(n + 4)
+ (3252704*n^5 + 2043740160 + 194208*n^6 + 2058817536*n + 33702144*n^4 +
221164160*n^3 + 897495552*n^2 + 6560*n^7 + 96*n^8)*a(n + 5) + (246432*n^6
+ 48931572*n^4 + 4055546880 + 1512709248*n^2 + 4406952*n^5 + 7824*n^7 +
345350856*n^3 + 108*n^8 + 3758813568*n)*a(n + 6) + (528439296*n + 2696360*n^4
+ 27036368*n^3 + 161115712*n^2 + 159784*n^5 + 5208*n^6 + 72*n^7 + 735989760)*a(n
+ 7) + ( - 59595808*n^2 - 8517816*n^3 - 338532480 - 504*n^6 - 680168*n^4
- 220837728*n - 28776*n^5)*a(n + 8) + ( - 262432*n^3 - 288*n^5 - 11355392*n
- 13824*n^4 - 20613120 - 2459328*n^2)*a(n + 9) + (31392*n^3 + 3713184*n
+ 720*n^4 + 512496*n^2 + 10074240)*a(n + 10) + (253440 + 288*n^3 + 8544*n^2
+ 82176*n)*a(n + 11) + ( - 7584*n - 49536 - 288*n^2)*a(n + 12) + 384*a(n + 13).
a(n) ~ n^(2*n) * 3^(n+1/2) / (exp(2*n+2) * 4^n). - Vaclav Kotesovec, Mar 11 2014
Recurrence (of order 11): 192*(243*n^2 - 285*n - 290)*a(n) = 144*(n-1)*(243*n^3 - 285*n^2 + 34*n + 796)*a(n-1) + 48*(n-2)*(n-1)*(1701*n^2 - 24*n + 1027)*a(n-2) - 48*(n-3)*(n-2)*(n-1)*(729*n^3 - 2556*n^2 - 2601*n + 2558)*a(n-3) + 8*(n-3)*(n-2)*(n-1)*(3645*n^3 - 7110*n^2 - 35091*n + 30676)*a(n-4) + 12*(n-4)*(n-3)*(n-2)*(n-1)*(729*n^4 - 4500*n^3 + 1623*n^2 + 12924*n - 11872)*a(n-5) + 8*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(729*n^3 - 4338*n^2 - 1728*n + 3269)*a(n-6) - 8*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(486*n^2 + 2145*n - 3485)*a(n-7) - 12*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(243*n^3 - 1014*n^2 - 1304*n + 1619)*a(n-8) + 24*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(18*n - 13)*a(n-9) - 6*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(27*n - 71)*a(n-10) + (n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(243*n^2 + 201*n - 332)*a(n-11). - Vaclav Kotesovec, Mar 11 2014

Extensions

Replaced broken link, Vaclav Kotesovec, Mar 11 2014

A110103 a(n) is the number of 2-regular 4-hypergraphs on 2n labeled vertices. (In a r-hypergraph, each hyper-edge is a proper r-set; k-regular implies that each vertex is in exactly k hyperedges.)

Original entry on oeis.org

1, 0, 0, 15, 1855, 469980, 214402650, 160081596675, 182667234224475, 302414315250247200, 697372026302486234700, 2167773244010692751057625, 8842276105055583472501844625, 46275602006744820263447546152500
Offset: 0

Views

Author

Marni Mishna, Jul 11 2005

Keywords

Comments

P-recursive.

Examples

			One of the 15 2-regular 4-hypergraphs on 6 vertices: {{1234},{4561}, {2356}}.

Links

Crossrefs

Cf. A025035, A110100, A110101, A025036.

Formula

Differential equation satisfied by exponential generating function sum a(n) t^(2n)/(2n)! {F(0) = 1, -144*t^3*(-2 + t^2)^2*(d^2/dt^2)F(t) - 12*(-2 + t^2)*(2*t^8-t^6 + 72 + 6*t^4-108*t^2)*(d/dt)F(t) - t^5*(-2 + t^2)*(t^2-3)*(t^4 + 4*t^2 + 36)*F(t)}.
Linear recurrence for a(n): {(15067980*n + 10550232*n^6 + 2859384*n^7 + 522720*n^8 + 128*n^11 + 2494800 + 61600*n^9 + 4224*n^10 + 52629038*n^3 + 45995730*n^4 + 26679070*n^5 + 37729494*n^2)*a(n) + (3791790*n + 109368*n^6 + 13872*n^7 + 1008*n^8 + 1247400 + 32*n^9 + 3747208*n^3 + 1767087*n^4 + 543858*n^5 + 4994577*n^2)*a(n + 1) + (28354500*n + 154560*n^6 + 11712*n^7 + 384*n^8 + 15478428*n^3 + 5309976*n^4 + 1152480*n^5 + 27874680*n^2 + 12474000)*a(n + 2) + (-623700-794025*n-48*n^6-115380*n^3-17760*n^4-1440*n^5-416757*n^2)*a(n + 3) + (599130*n + 534600 + 267282*n^2 + 59328*n^3 + 6552*n^4 + 288*n^5)*a(n + 4) + (-14166*n-26730-2484*n^2-144*n^3)*a(n + 5) + 54*a(n + 6), a(3) = 15, a(4) = 1855, a(5) = 469980, a(0) = 1, a(1) = 0, a(2) = 0}
Recurrence (of order 5): 54*(3*n - 4)*a(n) = 18*(n-1)*(2*n - 1)*(12*n^2 - 16*n + 9)*a(n-1) + 18*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(3*n + 1)*a(n-2) + 3*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(24*n^2 - 65*n + 24)*a(n-3) + 3*(n-3)*(n-2)*(n-1)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n + 1)*a(n-4) + 2*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n - 1)*a(n-5). - Vaclav Kotesovec, Mar 11 2014
a(n) ~ 2^(3*n+1) * n^(3*n) / (3^n * exp(3*n+3/2)). - Vaclav Kotesovec, Mar 11 2014

Extensions

Replaced broken link, Vaclav Kotesovec, Mar 11 2014
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