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

0,4

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

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.

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A218103 A289304 A281568 * A190567 A376235 A060339

Adjacent sequences: A084873 A084874 A084875 * A084877 A084878 A084879

Keyword

nonn

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Status

approved