A047707 -id:A047707 - cp's OEIS frontend

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

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

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..520
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

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

[(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

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 *)

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

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

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..415
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

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

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 *)

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).

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

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..345
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

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

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

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for linear recurrences with constant coefficients, signature (18,-99,162).

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

[(9^n - 2*6^n + 3*3^n)/2: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(9^n - 2*6^n + 3*3^n)/2, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((9^n - 2*6^n + 3*3^n)/2, ", ")) \\ G. C. Greubel, Oct 08 2017

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

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..520
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

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

[(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

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 *)

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

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

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

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.

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for linear recurrences with constant coefficients, signature (77,-2277,32895,-242514,854388,-1102248).

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

[(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

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 *)

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

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

cp's OEIS Frontend

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

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

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

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

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

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

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