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.

A134957 Number of hyperforests with n unlabeled vertices: analog of A134955 when edges of size 1 are allowed (with no two equal edges).

Original entry on oeis.org

1, 2, 6, 20, 75, 310, 1422, 7094, 37877, 213610, 1256422, 7641700, 47735075, 304766742, 1981348605, 13079643892, 87480944764, 591771554768, 4042991170169, 27864757592632, 193549452132550, 1353816898675732, 9529263306483357, 67457934248821368, 480019516988969011
Offset: 0

Views

Author

Don Knuth, Jan 26 2008

Keywords

Examples

			From _Gus Wiseman_, May 20 2018: (Start)
Non-isomorphic representatives of the a(3) = 20 hyperforests are the following:
  {}
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{1},{2}}
  {{1},{2,3}}
  {{2},{1,2}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{1},{2},{3}}
  {{1},{2},{1,2}}
  {{1},{3},{2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
(End)

Links

Crossrefs

Cf. A030019, A035053, A048143, A054921, A134955, A134957, A134959, A144959, A304867, A304911.

Programs

  • Mathematica
    etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];
b[n_] := Module[{v = {1}}, For[i = 2, i <= n, i++, v = Join[{1}, EulerT[EulerT[2 v]]]]; v];
seq[n_] := Module[{u = 2 b[n]}, Join[{1}, EulerT[ser[EulerT[u]]*(1 - x*ser[u]) + O[x]^n // CoefficientList[#, x]&]]];
seq[24] (* Jean-François Alcover, Feb 10 2020, after Andrew Howroyd *)
  • PARI
    \\ here b(n) is A318494 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
seq(n)={my(u=2*b(n)); concat([1], EulerT(Vec(Ser(EulerT(u))*(1-x*Ser(u)))))} \\ Andrew Howroyd, Aug 27 2018

Formula

Euler transform of A134959. - Gus Wiseman, May 20 2018

Extensions

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018

A134959 Number of spanning hypertrees with n unlabeled vertices: analog of A035053 when edges of size 1 are allowed (with no two equal edges).

Original entry on oeis.org

1, 2, 3, 10, 35, 150, 707, 3700, 20470, 119260, 719341, 4466316, 28367118, 183620874, 1207563011, 8049914664, 54295152117, 369981325578, 2544017965638, 17633790542978, 123108792874528, 865045359778662, 6114040341515978, 43443726772579152, 310195170229429300
Offset: 0

Views

Author

Don Knuth, Jan 26 2008

Keywords

Examples

			From _Gus Wiseman_, May 20 2018: (Start)
Non-isomorphic representatives of the a(3) = 10 hypertrees are the following:
  {{1,2,3}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,3},{2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}
(End)

Links

Crossrefs

Cf. A030019, A035053, A048143, A054921, A134955, A134957, A144959, A304867, A304911, A318494.

Programs

  • Mathematica
    etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
EulerT[v_List] := With[{q = etr[v[[#]] &]}, q /@ Range[Length[v]]];
ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];
b[n_] := Module[{v = {1}}, For[i = 2, i <= n, i++, v = Join[{1}, EulerT[EulerT[2 v]]]]; v];
seq[n_] := Module[{u = 2 b[n]}, 1 + x*ser[EulerT[u]]*(1 - x*ser[u]) + O[x]^n // CoefficientList[#, x]&];
seq[25] (* Jean-François Alcover, Feb 10 2020, after Andrew Howroyd *)
  • PARI
    \\ here b(n) is A318494 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
seq(n)={my(u=2*b(n)); Vec(1 + x*Ser(EulerT(u))*(1-x*Ser(u)))} \\ Andrew Howroyd, Aug 27 2018

Formula

Inverse Euler transform of A134957. - Gus Wiseman, May 20 2018

Extensions

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018

A304386 Number of unlabeled hypertrees (connected antichains with no cycles) spanning up to n vertices and allowing singleton edges.

Original entry on oeis.org

1, 2, 5, 15, 50, 200, 907, 4607, 25077, 144337, 863678, 5329994, 33697112, 217317986, 1424880997, 9474795661, 63769947778, 433751273356, 2977769238994, 20611559781972, 143720352656500, 1008765712435162, 7122806053951140, 50566532826530292, 360761703055959592
Offset: 0

Views

Author

Gus Wiseman, May 21 2018

Keywords

Examples

			Non-isomorphic representatives of the a(3) = 15 hypertrees are the following:
  {}
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{2},{1,2}}
  {{1,3},{2,3}}
  {{3},{1,2,3}}
  {{1},{2},{1,2}}
  {{3},{1,2},{2,3}}
  {{3},{1,3},{2,3}}
  {{2},{3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{2},{3},{1,2},{1,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}

Links

Crossrefs

Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A134959, A144959, A304386, A304717, A304867, A304911, A304912, A304968, A304970.

Programs

  • PARI
    \\ here b(n) is A318494 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
seq(n)={my(u=2*b(n)); Vec(1 + x*Ser(EulerT(u))*(1-x*Ser(u))/(1-x))} \\ Andrew Howroyd, Aug 27 2018

Formula

Partial sums of b(1) = 1, b(n) = A134959(n) otherwise.

Extensions

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018

A304977 Number of unlabeled hyperforests spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 14, 55, 235, 1112, 5672, 30783, 175733, 1042812, 6385278, 40093375, 257031667, 1676581863, 11098295287, 74401300872, 504290610004, 3451219615401, 23821766422463, 165684694539918, 1160267446543182, 8175446407807625, 57928670942338011, 412561582740147643
Offset: 0

Views

Author

Gus Wiseman, May 22 2018

Keywords

Examples

			Non-isomorphic representatives of the a(3) = 14 hyperforests are the following:
  {{1,2,3}}
  {{3},{1,2}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{1},{2},{3}}
  {{2},{3},{1,3}}
  {{2},{3},{1,2,3}}
  {{3},{1,2},{2,3}}
  {{3},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
  {{1},{2},{3},{1,2,3}}
  {{2},{3},{1,2},{1,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}

Links

Crossrefs

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304867, A304911, A304912, A304918.

Programs

  • PARI
    \\ here b(n) is A318494 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
seq(n)={my(u=2*b(n)); concat([1], EulerT(Vec(Ser(EulerT(u))*(1-x*Ser(u))-1)))} \\ Andrew Howroyd, Aug 27 2018

Formula

Euler transform of b(1) = 1, b(n > 1) = A134959(n).

Extensions

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018
Showing 1-4 of 4 results.

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