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.

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A084878 Number of (k,m,n)-antichains of multisets with k=3 and m=6.

Original entry on oeis.org

0, 0, 0, 15, 1729366, 10340309701, 24380294253318, 36539301527565851, 42407896071362952494, 42091311943805278602897, 37781049596189171124466966, 31727275407315883994852626087
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Comments

By a (k,m,n)-antichain of multisets we mean an m-antichain of k-bounded multisets on an n-set. 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.

Formula

a(n) = (1/6!)*(729^n - 30*486^n + 120*378^n + 60*324^n + 60*294^n - 360*279^n - 12*276^n - 720*252^n + 15*243^n + 90*234^n + 720*231^n + 120*216^n + 720*210^n - 240*205^n + 360*196^n - 720*189^n - 180*187^n + 720*186^n - 720*176^n + 120*168^n - 720*167^n + 360*165^n - 300*162^n - 720*157^n + 180*156^n + 720*148^n - 240*145^n + 720*138^n + 30*134^n - 240*129^n + 900*126^n - 360*120^n + 180*111^n + 300*108^n - 20*102^n + 150*98^n - 1800*93^n - 1800*84^n + 85*81^n + 450*78^n + 1800*77^n + 1800*70^n - 1800*63^n + 300*56^n - 1020*54^n + 2040*42^n + 340*36^n - 2040*31^n + 225*27^n + 510*26^n - 1350*18^n + 1350*14^n + 274*9^n - 548*6^n + 120*3^n).

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

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

A133789 Let P(A) denote the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, 1) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 2) x and y intersect but for which x is not a subset of y and y is not a subset of x.

Original entry on oeis.org

0, 1, 4, 16, 70, 316, 1414, 6196, 26590, 112156, 466774, 1923076, 7863310, 31972396, 129459334, 522571156, 2104535230, 8460991036, 33972711094, 136277478436, 546270602350, 2188566048076, 8764718254054, 35090241492916, 140455083984670, 562102715143516
Offset: 0

Views

Author

Ross La Haye, Jan 03 2008, Jan 08 2008

Keywords

Comments

Also, number of even binomial coefficient in rows 0 to 2^n of Pascal's triangle. [Aaron Meyerowitz, Oct 29 2013]

Examples

			a(3) = 16 because for P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} we see that
{1} and {2},
{1} and {3},
{2} and {3},
{1} and {2,3},
{2} and {1,3},
{3} and {1,2}
are disjoint, while
{} and {1},
{} and {2},
{} and {3},
{} and {1,2},
{} and {1,3},
{} and {2,3},
{} and {1,2,3}
are disjoint and one is a subset of the other and
{1,2} and {1,3},
{1,2} and {2,3},
{1,3} and {2,3}
are intersecting, but neither is a subset of the other.
Also, through row 8 of Pascal's triangle the a(3)=16 even entries are 2 (so a(0)=0 and a(1)=1) then 4,6,4 (so a(2)=4) then 10,10 then  6,20,6 then 8,28,56,70,56,28,8. [_Aaron Meyerowitz_, Oct 29 2013]

Links

Crossrefs

Cf. A082134, A002697, A020522, A006516, A007582, A000302.
Cf. A000225, A000392, A032263.

Formula

a(n) = (1/2)(4^n - 2*3^n + 3*2^n - 2).
O.g.f.: x*(1-6*x+11*x^2)/[(-1+x)*(-1+2*x)*(-1+3*x)*(-1+4*x)]. - R. J. Mathar, Jan 11 2008
a(n) = A084869(n)-1 = A016269(n-2)+2^n-1. - Vladeta Jovovic, Jan 04 2008, corrected by Eric Rowland, May 15 2017
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
a(n) = 10*a(n-1)-35*a(n-2)+50*a(n-3)-24*a(n-4). [Aaron Meyerowitz, Oct 29 2013]

Extensions

Edited by N. J. A. Sloane, Jan 20 2008 to incorporate suggestions from several contributors.
Previous Showing 11-15 of 15 results.

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