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 1-10 of 15 results. Next

A084883 Number of (k,m,n)-multiantichains of multisets with k=3 and m=6.

Original entry on oeis.org

1, 3, 64, 8022, 6822072, 14068794534, 26314469636622, 37310026340520678, 42667193588371160460, 42169580808988409450310, 37803058273249518925923210, 31733179110752959606870643334
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-A084882.

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 + 45*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 - 900*162^n - 720*157^n + 180*156^n + 720*148^n - 240*145^n + 720*138^n + 30*134^n - 240*129^n + 2700*126^n - 360*120^n + 180*111^n + 900*108^n - 20*102^n + 450*98^n - 5400*93^n - 5400*84^n + 685*81^n + 1350*78^n + 5400*77^n + 5400*70^n - 5400*63^n + 900*56^n - 8220*54^n + 16440*42^n + 2740*36^n - 16440*31^n + 4275*27^n + 4110*26^n - 25650*18^n + 25650*14^n + 10474*9^n - 20948*6^n + 7560*3^n).

A084870 Number of 3-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 6, 28, 190, 1692, 16766, 166028, 1586430, 14580412, 129654526, 1123451628, 9544185470, 79881877532, 661135445886, 5425962250828, 44250287565310, 359161631645052, 2904756409742846, 23429320590259628, 188594431902253950
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
  • PARI
    for(n=0,50, print1((8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n)/6, ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (1/3!)*(8^n - 6*6^n + 6*5^n + 9*4^n - 18*3^n + 14*2^n).
G.f.: ( 1-26*x+265*x^2-1330*x^3+3340*x^4-3432*x^5 ) / ( (6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(8*x-1)*(5*x-1) ). - R. J. Mathar, Jul 08 2011

A084882 Number of (k,m,n)-multiantichains of multisets with k=3 and m=5.

Original entry on oeis.org

1, 3, 51, 4129, 1439381, 814788851, 395927618035, 155157302244381, 51960586962031617, 15663181302847575559, 4402571746033946222639, 1180812802393866826858193, 306839347397532891662028733
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

  • Mathematica
    Table[(1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 30*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 360*54^n + 720*42^n + 120*36^n - 720*31^n + 275*27^n + 180*26^n - 1650*18^n + 1650*14^n + 870*9^n - 1740*6^n + 744*3^n), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

Formula

a(n) = (1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 30*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 360*54^n + 720*42^n + 120*36^n - 720*31^n + 275*27^n + 180*26^n - 1650*18^n + 1650*14^n + 870*9^n - 1740*6^n + 744*3^n).

A084871 Number of 4-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 7, 41, 398, 6177, 128232, 2881531, 62769238, 1288737197, 25012685732, 463681018671, 8294783320578, 144410750517217, 2462999084589232, 41359616334934211, 686406989350511918, 11290725888842193237
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [(16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/24: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[(16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/4!, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
  • PARI
    for(n=0,50, print1((16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/4!, ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (16^n - 12*12^n + 24*10^n + 4*9^n - 6*8^n + 6*7^n - 108*6^n + 108*5^n + 83*4^n - 166*3^n + 90*2^n)/4!.
From R. J. Mathar, Jul 08 2011: (Start)
G.f.: (-1 + 80*x - 2813*x^2 + 57293*x^3 - 749139*x^4 + 6577949*x^5 - 39353597*x^6 + 158972472*x^7 - 417774220*x^8 + 651991536*x^9 - 465379200*x^10) / ( (9*x-1) *(6*x-1) *(7*x-1) *(3*x-1) *(5*x-1) *(2*x-1) *(12*x-1) *(10*x-1) *(4*x-1) *(8*x-1) *(16*x-1) ).
a(n) = 82*a(n-1) - 2970*a(n-2) + 62700*a(n-3) - 856713*a(n-4) + 7947786*a(n-5) - 51019100*a(n-6) + 226259000*a(n-7) - 678011136*a(n-8) + 1304341632*a(n-9) - 1445575680*a(n-10) + 696729600*a(n-11). (End)

A084872 Number of 5-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 8, 56, 726, 17938, 722680, 35955180, 1798971434, 83885891894, 3612380896332, 145277787750064, 5534505187364062, 202229611397865690, 7158136006402746464, 247316732670273773108, 8389241054998193347410
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [(32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n)/120: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[(32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n)/120, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
  • PARI
    for(n=0,50, print1((32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n)/120, ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (1/5!)*(32^n - 20*24^n + 60*20^n + 20*18^n + 10*17^n - 90*16^n - 120*15^n + 150*14^n + 120*13^n - 480*12^n + 20*11^n + 720*10^n + 120*9^n - 445*8^n + 180*7^n - 1650*6^n + 1650*5^n + 870*4^n - 1740*3^n + 744*2^n).

A084873 Number of 6-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 9, 73, 1212, 44667, 3251186, 345094227, 39552733796, 4234657495267, 409948262617398, 36190736880911571, 2964860272283578040, 229165985114590010307, 16940021231116707830570
Offset: 0

Views

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    Table[(1/6!)*(64^n - 30*48^n + 120*40^n + 60*36^n + 60*34^n - 12*33^n - 315*32^n - 720*30^n + 810*28^n + 120*27^n + 480*26^n + 360*25^n - 1080*24^n - 720*23^n - 240*22^n - 540*21^n + 3180*20^n + 750*19^n + 660*18^n + 90*17^n - 4535*16^n - 5420*15^n + 6750*14^n + 5400*13^n - 13620*12^n + 900*11^n + 16440*10^n + 2740*9^n - 12165*8^n + 4110*7^n - 25650*6^n + 25650*5^n + 10474*4^n - 20948*3^n + 7560*2^n), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

Formula

a(n) = (1/6!)*(64^n - 30*48^n + 120*40^n + 60*36^n + 60*34^n - 12*33^n - 315*32^n - 720*30^n + 810*28^n + 120*27^n + 480*26^n + 360*25^n - 1080*24^n - 720*23^n - 240*22^n - 540*21^n + 3180*20^n + 750*19^n + 660*18^n + 90*17^n - 4535*16^n - 5420*15^n + 6750*14^n + 5400*13^n - 13620*12^n + 900*11^n + 16440*10^n + 2740*9^n - 12165*8^n + 4110*7^n - 25650*6^n + 25650*5^n + 10474*4^n - 20948*3^n + 7560*2^n).

A084874 Number of (k,m,n)-antichains of multisets with k=3 and m=2.

Original entry on oeis.org

0, 0, 9, 162, 2025, 21870, 219429, 2112642, 19847025, 183642390, 1682955549, 15327821322, 139038251625, 1257873017310, 11360034454869, 102475388237202, 923689006041825, 8321664254958630, 74945757885541389, 674816499677616282
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.
a(n) is also the number of entries that are divisible by 3 in rows 0 through 3^n-1 of Pascal's triangle A007318. - Tim Cieplowski, Nov 25 2014

Links

Crossrefs

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

Programs

  • Magma
    [(9^n - 2*6^n + 3^n)/2: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[(9^n - 2*6^n + 3^n)/2, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
LinearRecurrence[{18,-99,162},{0,0,9},20] (* Harvey P. Dale, Oct 01 2023 *)
  • PARI
    for(n=0,50, print1((9^n - 2*6^n + 3^n)/2, ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (1/2!)*(9^n - 2*6^n + 3^n).
G.f.: -9*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) + exp(3*x))/2. - G. C. Greubel, Oct 08 2017

A084875 Number of (k,m,n)-antichains of multisets with k=3 and m=3.

Original entry on oeis.org

0, 0, 1, 350, 24025, 1061570, 38306701, 1238697950, 37547263825, 1093418309690, 31035659056501, 866306577308150, 23915774118612025, 655397866616830610, 17872808187862527901, 485794481046271815950, 13175146525408965630625
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.

Programs

  • Magma
    [(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)
LinearRecurrence[{77,-2277,32895,-242514,854388,-1102248},{0,0,1,350,24025,1061570},20] (* Harvey P. Dale, May 29 2025 *)
  • PARI
    for(n=0,50, print1((27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6, ", ")) \\ G. C. Greubel, Oct 08 2017

Formula

a(n) = (1/3!)*(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n).
G.f.: -x^2*(-1-273*x+648*x^2+24300*x^3) / ( (18*x-1)*(9*x-1)*(6*x-1)*(3*x-1)*(14*x-1)*(27*x-1) ). - R. J. Mathar, Jul 08 2011

A084876 Number of (k,m,n)-antichains of multisets with k=3 and m=4.

Original entry on oeis.org

0, 0, 0, 310, 159300, 32389900, 4469327850, 503689260970, 50466655894200, 4701945998612200, 418104908350395750, 36055756736065208230, 3046399249526576159700, 253883533322134812268900, 20963248884482293139928450, 1720141562616331422239725090
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.

Programs

  • Magma
    [(81^n - 12*54^n + 24*42^n + 4*36^n - 24*31^n + 6*27^n + 6*26^n - 36*18^n + 36*14^n + 11*9^n - 22*6^n + 6*3^n)/24: n in [0..50]]; // G. C. Greubel, Oct 08 2017
  • Mathematica
    Table[(81^n - 12*54^n + 24*42^n + 4*36^n - 24*31^n + 6*27^n + 6*26^n - 36*18^n + 36*14^n + 11*9^n - 22*6^n + 6*3^n)/24, {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 + 6*27^n + 6*26^n - 36*18^n + 36*14^n + 11*9^n - 22*6^n + 6*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 + 6*27^n + 6*26^n - 36*18^n + 36*14^n + 11*9^n - 22*6^n + 6*3^n).
G.f.: -10*x^3*(-31 - 5173*x + 663390*x^2 - 16812297*x^3 - 320866029*x^4 + 19383439320*x^5 - 243502067160*x^6 + 252158125680*x^7 + 6816687418800*x^8) / ( (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

A084877 Number of (k,m,n)-antichains of multisets with k=3 and m=5.

Original entry on oeis.org

0, 0, 0, 114, 649850, 678772108, 377819587984, 153135104560046, 51758494975477206, 15644366957608679376, 4400899140179858419388, 1180668574169021790713938, 306827161657039584492179842
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.

Programs

  • Mathematica
    Table[(1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 10*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 120*54^n + 240*42^n + 40*36^n - 240*31^n + 35*27^n + 60*26^n - 210*18^n + 210*14^n + 50*9^n - 100*6^n + 24*3^n), {n, 0, 1000}] (* G. C. Greubel, Oct 08 2017 *)

Formula

a(n) = (1/5!)*(243^n - 20*162^n + 60*126^n + 20*108^n + 10*98^n - 120*93^n - 120*84^n + 10*81^n + 30*78^n + 120*77^n + 120*70^n - 120*63^n + 20*56^n - 120*54^n + 240*42^n + 40*36^n - 240*31^n + 35*27^n + 60*26^n - 210*18^n + 210*14^n + 50*9^n - 100*6^n + 24*3^n).
Showing 1-10 of 15 results. Next

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