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).

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

Author

Venkat Murali (v.murali(AT)ru.ac.za), May 28 2005

Keywords

• nice
• nonn
• easy

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.

Examples

			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.
		

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

• Colin Barker, Table of n, a(n) for n = 0..1000
• Venkat Murali, Home page.
• Index entries for linear recurrences with constant coefficients, signature (9,-32,56,-48,16).

Crossrefs

Cf. A107464, A007047, A107392.

Programs

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 *)

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

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

Extensions

a(5) corrected Jun 01 2005

Incorrect term deleted by Colin Barker, Jan 15 2015