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%I #14 Feb 05 2023 09:18:05
%S 31,175,703,2415,7551,22143,61951,167167,438271,1122303,2818047,
%T 6959103,16941055,40730623,96862207,228130815,532676607,1234173951,
%U 2839543807,6491734015,14755561471,33361494015,75061264383,168124481535,375004332031,833223655423
%N Number of chains in the power set lattice of an (n+3)-element set X_(n+3) of specification n^1 2^1 1, that is, n identical objects of one kind, 2 identical objects of another kind and one other kind. It is the same as the number of fuzzy subsets X_(n+3).
%C This sequence is one of a triple sequence A(n,m,l) of the number of fuzzy subsets of a set with n+m+l objects of 3 kinds. There are n,m and l objects for each kind respectively. Here a(n)= A(n,2,1). The sequence A107464 is one other example of A(n,m,l) for m=l=1.
%D V. Murali, On the number of fuzzy subsets of an (n+3)-element set of specification n^1 2^1 1, Rhodes University Preprint, 2005.
%H Colin Barker, <a href="/A107953/b107953.txt">Table of n, a(n) for n = 0..1000</a>
%H Venkat Murali, <a href="https://www.ru.ac.za/mathematics/people/staff/venkatmurali/">Home page</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-32,56,-48,16).
%F a(n) = 2^(n+1) * ((n^2/2)*(n+13) + 21*n + 16) - 1.
%F G.f.: (48*x^3-120*x^2+104*x-31) / ((x-1)*(2*x-1)^4). - _Colin Barker_, Jan 15 2015
%F a(0)=31, a(1)=175, a(2)=703, a(3)=2415, a(4)=7551, a(n)=9*a(n-1)- 32*a(n-2)+ 56*a(n-3)-48*a(n-4)+16*a(n-5). - _Harvey P. Dale_, Feb 10 2015
%e a(3) = 2^4*((9/2)*16 + 21*3 + 16) - 1 = 2415 which is the number of distinct chains in the power set lattice (or fuzzy subsets) of a set X_(n+3) with 3 kinds of objects, n of one kind, 2 of another and one of yet another.
%t Table[2^(n+1) (n^2/2 (n+13)+21n+16)-1,{n,0,30}] (* or *) LinearRecurrence[ {9,-32,56,-48,16},{31,175,703,2415,7551},30] (* _Harvey P. Dale_, Feb 10 2015 *)
%o (PARI) Vec((48*x^3-120*x^2+104*x-31)/((x-1)*(2*x-1)^4) + O(x^100)) \\ _Colin Barker_, Jan 15 2015
%Y Cf. A107464, A007047, A107392.
%K nice,nonn,easy
%O 0,1
%A Venkat Murali (v.murali(AT)ru.ac.za), May 28 2005
%E a(5) corrected Jun 01 2005
%E Incorrect term deleted by _Colin Barker_, Jan 15 2015
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