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.

Previous Showing 11-18 of 18 results.

A304918 Number of labeled antichain hyperforests spanning a subset of {1,...,n}.

Original entry on oeis.org

1, 2, 5, 18, 104, 943, 12133, 203038, 4177755, 101922814, 2874725600, 92009680557, 3294276613933, 130446181101044, 5660055256165565, 267044522107706072, 13611243187516647324, 745329728016955480687, 43636132793651444511809, 2719977663069107176768790
Offset: 0

Views

Author

Gus Wiseman, May 21 2018

Keywords

Examples

			The a(3) = 18 hyperforests are the following:
{{1,2,3}}      {{2,3}}    {{1,3}}    {{1,2}}    {{3}}   {{2}}   {{1}}   {}
{{1,3},{2,3}}  {{2},{3}}  {{1},{3}}  {{1},{2}}
{{1,2},{2,3}}
{{1,2},{1,3}}
{{3},{1,2}}
{{2},{1,3}}
{{1},{2,3}}
{{1},{2},{3}}

Crossrefs

Cf. A030019, A035053, A048143, A134954, A134955, A134957, A134956, A134957, A134959, A144959, A303838, A304716, A304717, A304867, A304911, A304912.

Formula

Binomial transform of A134954.

A304968 Number of labeled hypertrees spanning some subset of {1,...,n}, with singleton edges allowed.

Original entry on oeis.org

1, 2, 7, 48, 621, 12638, 351987, 12426060, 531225945, 26674100154, 1538781595999, 100292956964456, 7288903575373509, 584454485844541718, 51256293341752583499, 4880654469385955209092, 501471626403154217825457, 55300894427785157597436786
Offset: 0

Views

Author

Gus Wiseman, May 22 2018

Keywords

Examples

			The a(2) = 7 hypertrees are the following:
{}
{{1}}
{{2}}
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Links

Crossrefs

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304386, A304867, A304911, A304912, A304918, A304968, A304970.

Programs

  • PARI
    \\ here b(n) is A134958 with b(1)=1.
b(n)=if(n<2, n>=0, 2^n*sum(i=0, n, stirling(n-1, i, 2)*n^(i-1)));
a(n)=sum(k=0, n, binomial(n, k)*b(k)); \\ Andrew Howroyd, Aug 27 2018

Formula

Binomial transform of b(1) = 1, b(n) = A134958(n) otherwise.

A304970 Number of unlabeled hypertrees with up to n vertices and without singleton edges.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 39, 98, 263, 759, 2299, 7259, 23649, 79057, 269629, 935328, 3290260, 11714285, 42139053, 152963037, 559697097, 2062574000, 7649550572, 28534096988, 106994891146, 403119433266, 1525466082179, 5795853930652, 22102635416716, 84579153865570
Offset: 0

Views

Author

Gus Wiseman, May 22 2018

Keywords

Examples

			Non-isomorphic representatives of the a(4) = 8 hypertrees are the following:
{}
{{1,2}}
{{1,2,3}}
{{1,2,3,4}}
{{1,3},{2,3}}
{{1,4},{2,3,4}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}

Links

Crossrefs

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304386, A304867, A304911, A304912, A304918, A304968, A304970.

Programs

  • PARI
    \\ here b(n) is A007563 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(v)))); v}
seq(n)={my(u=b(n)); Vec(1 + (x*Ser(EulerT(u))*(1-x*Ser(u)))/(1-x))} \\ Andrew Howroyd, Aug 27 2018

Formula

Partial sums of A035053 if we assume A035053(1) = 0.
a(n) = A304937(n) + 1 for n > 0.

A305028 Number of unlabeled blobs spanning n vertices without singleton edges.

Original entry on oeis.org

1, 0, 1, 2, 10, 128
Offset: 0

Views

Author

Gus Wiseman, May 24 2018

Keywords

Comments

A blob is a connected antichain of finite sets that cannot be capped by a hypertree with more than one branch.

Examples

			Non-isomorphic representatives of the a(4) = 10 blobs:
  {{1,2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Links

Crossrefs

Cf. A030019, A035053, A048143, A054921, A134957, A144959, A275307, A286520, A293607, A304118, A304717, A304867.

A304937 Number of unlabeled nonempty hypertrees with up to n vertices and no singleton edges.

Original entry on oeis.org

1, 0, 1, 3, 7, 16, 38, 97, 262, 758, 2298, 7258, 23648, 79056, 269628, 935327, 3290259, 11714284, 42139052, 152963036, 559697096, 2062573999, 7649550571, 28534096987, 106994891145, 403119433265, 1525466082178, 5795853930651, 22102635416715, 84579153865569
Offset: 0

Views

Author

Gus Wiseman, May 21 2018

Keywords

Examples

			Non-isomorphic representatives of the a(5) = 16 hypertrees are the following:
{{1,2}}
{{1,2,3}}
{{1,2,3,4}}
{{1,2,3,4,5}}
{{1,3},{2,3}}
{{1,4},{2,3,4}}
{{1,5},{2,3,4,5}}
{{1,2,5},{3,4,5}}
{{1,2},{2,5},{3,4,5}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}
{{1,4},{2,5},{3,4,5}}
{{1,5},{2,5},{3,4,5}}
{{1,3},{2,4},{3,5},{4,5}}
{{1,4},{2,5},{3,5},{4,5}}
{{1,5},{2,5},{3,5},{4,5}}

Links

Crossrefs

Cf. A030019, A035053, A048143, A134954, A134955, A134956, A134957, A134959, A144959, A303838, A304867, A304911, A304912, A304918.

Programs

  • PARI
    \\ here b(n) is A007563 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(v)))); v}
seq(n)={my(u=b(n)); Vec(1 + (x*Ser(EulerT(u))*(1-x*Ser(u)) - x)/(1-x))} \\ Andrew Howroyd, Aug 27 2018

Formula

a(n) = a(n-1) + A035053(n) for n > 1, a(n) = 1 - n for n < 2.

A304939 Number of labeled nonempty hypertrees (connected antichains with no cycles) spanning some subset of {1,...,n} without singleton edges.

Original entry on oeis.org

1, 0, 1, 7, 51, 506, 6843, 118581, 2504855, 62370529, 1788082153, 57997339632, 2099638691439, 83922479506503, 3670657248913385, 174387350448735877, 8942472292255441103, 492294103555090048458, 28958704109012732921523
Offset: 0

Views

Author

Gus Wiseman, May 21 2018

Keywords

Examples

			The a(3) = 7 hypertrees are the following:
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}

Links

Crossrefs

Cf. A030019, A035053, A048143, A134954, A134955, A134956, A134957, A134959, A144959, A303838, A304867, A304911, A304912, A304918, A305004.

Programs

  • PARI
    \\ here b(n) is A030019 with b(1)=0.
b(n)=if(n<2, n==0, sum(i=0, n, stirling(n-1, i, 2)*n^(i-1)));
a(n)=if(n<1, n==0, sum(k=1, n, binomial(n, k)*b(k))); \\ Andrew Howroyd, Aug 27 2018

Formula

a(n) = A305004(n) - 1 for n > 0. - 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

A304919 Number of labeled hyperforests spanning {1,...,n} and allowing singleton edges.

Original entry on oeis.org

1, 1, 5, 45, 665, 14153, 399421, 14137301, 603647601, 30231588689, 1738713049013, 112976375651901, 8186616300733321, 654642360222892057, 57267075701210437229, 5440407421313402397541, 557802495215406348358113, 61393838258161429159571873, 7220049654850517272144419941, 903546142463635579042416518989
Offset: 0

Views

Author

Gus Wiseman, May 21 2018

Keywords

Examples

			The a(2) = 5 hyperforests are the following:
{{1,2}}
{{1},{2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Crossrefs

Cf. A030019, A035053, A048143, A134954, A134955, A134956, A134957, A134959, A144959, A303838, A304717, A304867, A304911, A304912, A304918.

Formula

Inverse binomial transform of A134956.
Previous Showing 11-18 of 18 results.

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

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