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%I #25 Oct 01 2023 11:38:05
%S 0,0,9,162,2025,21870,219429,2112642,19847025,183642390,1682955549,
%T 15327821322,139038251625,1257873017310,11360034454869,
%U 102475388237202,923689006041825,8321664254958630,74945757885541389,674816499677616282
%N Number of (k,m,n)-antichains of multisets with k=3 and m=2.
%C 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.
%C 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
%H G. C. Greubel, <a href="/A084874/b084874.txt">Table of n, a(n) for n = 0..1000</a>
%H Goran Kilibarda and Vladeta Jovovic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kilibarda/kili2.html">Antichains of Multisets</a>, J. Integer Seqs., Vol. 7, 2004.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (18,-99,162).
%F a(n) = (1/2!)*(9^n - 2*6^n + 3^n).
%F G.f.: -9*x^2 / ( (6*x-1)*(3*x-1)*(9*x-1) ). - _R. J. Mathar_, Jul 08 2011
%F E.g.f.: (exp(9*x) - 2*exp(6*x) + exp(3*x))/2. - _G. C. Greubel_, Oct 08 2017
%t Table[(9^n - 2*6^n + 3^n)/2, {n, 0, 50}] (* _G. C. Greubel_, Oct 08 2017 *)
%t LinearRecurrence[{18,-99,162},{0,0,9},20] (* _Harvey P. Dale_, Oct 01 2023 *)
%o (PARI) for(n=0,50, print1((9^n - 2*6^n + 3^n)/2, ", ")) \\ _G. C. Greubel_, Oct 08 2017
%o (Magma) [(9^n - 2*6^n + 3^n)/2: n in [0..50]]; // _G. C. Greubel_, Oct 08 2017
%Y Cf. A016269, A047707, A051112-A051118, A084869-A084883.
%K nonn
%O 0,3
%A Goran Kilibarda, _Vladeta Jovovic_, Jun 10 2003
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