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

0, 0, 9, 162, 2025, 21870, 219429, 2112642, 19847025, 183642390, 1682955549, 15327821322, 139038251625, 1257873017310, 11360034454869, 102475388237202, 923689006041825, 8321664254958630, 74945757885541389, 674816499677616282

Offset: 0

Author

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

Keywords

• nonn

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

• 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).

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