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

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

Offset: 0

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

• nonn

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

• 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.

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