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A084875 Number of (k,m,n)-antichains of multisets with k=3 and m=3.

%I A084875 #22 May 29 2025 12:01:20
%S A084875 0,0,1,350,24025,1061570,38306701,1238697950,37547263825,
%T A084875 1093418309690,31035659056501,866306577308150,23915774118612025,
%U A084875 655397866616830610,17872808187862527901,485794481046271815950,13175146525408965630625
%N A084875 Number of (k,m,n)-antichains of multisets with k=3 and m=3.
%C A084875 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.
%H A084875 G. C. Greubel, <a href="/A084875/b084875.txt">Table of n, a(n) for n = 0..695</a>
%H A084875 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 A084875 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (77,-2277,32895,-242514,854388,-1102248).
%F A084875 a(n) = (1/3!)*(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n).
%F A084875 G.f.: -x^2*(-1-273*x+648*x^2+24300*x^3) / ( (18*x-1)*(9*x-1)*(6*x-1)*(3*x-1)*(14*x-1)*(27*x-1) ). - _R. J. Mathar_, Jul 08 2011
%t A084875 Table[(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6, {n, 0, 50}] (* _G. C. Greubel_, Oct 08 2017 *)
%t A084875 LinearRecurrence[{77,-2277,32895,-242514,854388,-1102248},{0,0,1,350,24025,1061570},20] (* _Harvey P. Dale_, May 29 2025 *)
%o A084875 (PARI) for(n=0,50, print1((27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6, ", ")) \\ _G. C. Greubel_, Oct 08 2017
%o A084875 (Magma) [(27^n - 6*18^n + 6*14^n + 3*9^n - 6*6^n + 2*3^n)/6: n in [0..50]]; // _G. C. Greubel_, Oct 08 2017
%Y A084875 Cf. A016269, A047707, A051112-A051118, A084869-A084883.
%K A084875 nonn
%O A084875 0,4
%A A084875 Goran Kilibarda, _Vladeta Jovovic_, Jun 10 2003