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 31-34 of 34 results.

A084879 Number of (k,m,n)-multiantichains of multisets with k=3 and m=2.

Original entry on oeis.org

1, 3, 18, 189, 2106, 22113, 220158, 2114829, 19853586, 183662073, 1683014598, 15327998469, 139038783066, 1257874611633, 11360039237838, 102475402586109, 923689049088546, 8321664384098793, 74945758272961878, 674816500839877749
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Comments

By a (k,m,n)-multiantichain of multisets we mean an m-multiantichain of k-bounded multisets on an n-set. The elements of a multiantichain could have the multiplicities greater than 1. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

Links

Crossrefs

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

Programs

  • Magma
    [(9^n - 2*6^n + 3*3^n)/2: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[(9^n - 2*6^n + 3*3^n)/2, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
  • PARI
    for(n=0,50, print1((9^n - 2*6^n + 3*3^n)/2, ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (9^n - 2*6^n + 3*3^n)/2.
G.f.: ( -1 + 15*x - 63*x^2 ) / ( (6*x-1)*(3*x-1)*(9*x-1) ). - R. J. Mathar, Jul 08 2011
E.g.f.: (exp(9*x) - 2*exp(6*x) + 3*exp(3*x))/2. - G. C. Greubel, Oct 08 2017

A084881 Number of (k,m,n)-multiantichains of multisets with k=3 and m=4.

Original entry on oeis.org

1, 3, 39, 1873, 237531, 35640463, 4584906969, 507411694933, 50579357233311, 4705226804488123, 418198020376490949, 36058355701780773793, 3046470997266047282091, 253885499519508283406983
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Comments

By a (k,m,n)-multiantichain of multisets we mean an m-multiantichain of k-bounded multisets on an n-set. The elements of a multiantichain could have the multiplicities greater than 1. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

Links

Crossrefs

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

Programs

  • Magma
    [(81^n - 12*54^n + 24*42^n + 4*36^n - 24*31^n + 18*27^n + 6*26^n - 108*18^n + 108*14^n + 83*9^n - 166*6^n + 90*3^n)/24: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[(1/4!)*(81^n - 12*54^n + 24*42^n + 4*36^n - 24*31^n + 18*27^n + 6*26^n - 108*18^n + 108*14^n + 83*9^n - 166*6^n + 90*3^n), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
  • PARI
    for(n=0,50, print1((81^n - 12*54^n + 24*42^n + 4*36^n - 24*31^n + 18*27^n + 6*26^n - 108*18^n + 108*14^n + 83*9^n - 166*6^n + 90*3^n)/24, ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (1/4!)*(81^n - 12*54^n + 24*42^n + 4*36^n - 24*31^n + 18*27^n + 6*26^n - 108*18^n + 108*14^n + 83*9^n - 166*6^n + 90*3^n).
G.f.: ( 1 - 344*x + 51428*x^2 - 4415688*x^3 + 242115073*x^4 - 8897167926*x^5 + 223317141174*x^6 - 3827454303870*x^7 + 44109912725856*x^8 - 331501702734000*x^9 + 1522496648595168*x^10 - 3394508914171872*x^11 ) / ( (6*x-1) *(54*x-1) *(42*x-1) * (3*x-1) *(9*x-1) *(27*x-1) *(31*x-1) *(26*x-1) *(18*x-1) *(81*x-1) *(36*x-1) *(14*x-1) ). - R. J. Mathar, Jul 08 2011

A056164 Number of ordered antichain covers of an unlabeled n-set; labeled T_1-hypergraphs (without empty hyperedges) with n hyperedges.

Original entry on oeis.org

1, 2, 6, 109, 191177
Offset: 1

Views

Author

Vladeta Jovovic, Goran Kilibarda, Jul 31 2000

Keywords

Comments

A T_1-hypergraph is a hypergraph (not necessarily without empty hyperedges or multiple hyperedges) which for every ordered pair of distinct nodes has a hyperedge containing one but not the other node.

Examples

			There are 6 ordered antichain covers on an unlabeled 3-set: ({1,2,3}), ({1},{2,3}), ({2,3},{1}), ({1,2},{1,3}), ({1},{2},{3}), ({1,2},{1,3},{2,3}).
a(3)=1+3+2=6; a(4)=1+6+17+25+30+30=109; a(5)=1+10+71+429+2176+8310+20580+38640+60480+60480=191177.

References

  • V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6)
  • V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

Links

Crossrefs

Cf. A056074, A056090, A056093, A000372, A056005, A056069-A056071, A056073, A056046-A056049, A056052, A056101, A056104, A051112-A051118.

Formula

a(n)=Sum_{k=1..C(n, floor(n/2))}b(k, n) where b(k, n) is the number of k-element ordered antichains covers of an unlabeled n-set.

A084880 Number of (k,m,n)-multiantichains of multisets with k=3 and m=3.

Original entry on oeis.org

1, 3, 28, 701, 28156, 1105553, 38746288, 1242925421, 37586964436, 1093785614153, 31039025026648, 866337233127941, 23916052195646716, 655400382364459553, 17872830907936220608, 485794685997062639261, 13175148372787020760996
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Comments

By a (k,m,n)-multiantichain of multisets we mean an m-multiantichain of k-bounded multisets on an n-set. The elements of a multiantichain could have the multiplicities greater than 1. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

Links

Crossrefs

Cf. A016269, A047707, A051112-A051118, A084869-A084883.

Programs

  • Magma
    [(27^n - 6*18^n + 6*14^n + 9*9^n - 18*6^n + 14*3^n)/6: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    LinearRecurrence[{77,-2277,32895,-242514,854388,-1102248},{1,3,28,701, 28156,1105553},20] (* Harvey P. Dale, Apr 08 2015 *)
Table[(27^n - 6*18^n + 6*14^n + 9*9^n - 18*6^n + 14*3^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
  • PARI
    for(n=0,50, print1((27^n - 6*18^n + 6*14^n + 9*9^n - 18*6^n + 14*3^n)/6, ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (27^n - 6*18^n + 6*14^n + 9*9^n - 18*6^n + 14*3^n)/6.
G.f.: (1 - 74*x + 2074*x^2 - 27519*x^3 + 181764*x^4 - 514188*x^5) / ( (18*x-1)*(9*x-1)*(6*x-1)*(3*x-1)*(14*x-1)*(27*x-1) ). - R. J. Mathar, Jul 08 2011
a(0)=1, a(1)=3, a(2)=28, a(3)=701, a(4)=28156, a(5)=1105553, a(n) = 77*a(n-1) - 2277*a(n-2) + 32895*a(n-3) - 242514*a(n-4) + 854388*a(n-5) - 1102248*a(n-6). - Harvey P. Dale, Apr 08 2015
Previous Showing 31-34 of 34 results.

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