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A107953
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).
3
31, 175, 703, 2415, 7551, 22143, 61951, 167167, 438271, 1122303, 2818047, 6959103, 16941055, 40730623, 96862207, 228130815, 532676607, 1234173951, 2839543807, 6491734015, 14755561471, 33361494015, 75061264383, 168124481535, 375004332031, 833223655423
OFFSET
0,1
COMMENTS
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.
REFERENCES
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.
LINKS
FORMULA
a(n) = 2^(n+1) * ((n^2/2)*(n+13) + 21*n + 16) - 1.
G.f.: (48*x^3-120*x^2+104*x-31) / ((x-1)*(2*x-1)^4). - Colin Barker, Jan 15 2015
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
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nice,nonn,easy
AUTHOR
Venkat Murali (v.murali(AT)ru.ac.za), May 28 2005
EXTENSIONS
a(5) corrected Jun 01 2005
Incorrect term deleted by Colin Barker, Jan 15 2015
STATUS
approved