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Search: id:A120672
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| A120672 |
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Consider a set A containing at least n-1 elements of sort "a" and a set B containing at least n-1 elements of sort "b". From set A we take i elements, from set B we take (n-i) elements such that i + (n-i) = n. Then we distribute these n elements in two urns L (left) and R (right). The order of selection among the two sorts counts. Equivalently we can say: Then we form two sequences L and R from these n elements. The position of the sort of the elements within the sequences counts. Furthermore, the occupations of the urns are permuted. In other words, the order of the sequences L and R is swapped from L|R to R|L. For n=3 we have a(n=3)=12 configurations [L|R] and [R|L]: [aaa|b], [b|aaa], [baa|a], [a|baa], [aba|a], [a|aba], [aab|a], [a|aab].and [bbb|a], [a|bbb], [abb|b], [b|abb], [bab|b], [b|bab], [bba|b], [b|bba]. |
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| 0, 0, 2, 12, 22, 60, 104, 252, 438, 1020, 1792, 4092, 7264, 16380, 29332, 65532, 118198, 262140, 475664, 1048572, 1912392, 4194300, 7683172
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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A028399(n) = 2*2^n-4 with n=1,2,3,... is an upper limit for A120672 because Sum(2*n!/i!/(n-i)!,i = 1 .. n-1) = 2*2^n-4. We have for A028399(n): 4,12,28,60,124,252,508,1020,2044,4092,8188,16380,32764,65532,131068,262140,524284,1048572,2097148, 4194300,8388604. A120672 follows from all distinct ordered 2-tuples of positive integers whose elements sum to n. See the following Maple program: A120672 := proc(n::integer) local i,k,cmpstnlst,cmpstn,NumberOfParts,liste,NumberOfDifferentParts,Result; k:=2; Result := 0; cmpstnlst := composition(n,k); NumberOfParts := 0; NumberOfDifferentParts := 0; for i from 1 to nops(cmpstnlst) do cmpstn := cmpstnlst[i]; NumberOfParts := nops(cmpstn); if NumberOfParts > 0 then liste := convert(cmpstn,multiset); else liste := NULL; fi; if liste <> NULL then NumberOfDifferentParts := nops(liste); else NumberOfDifferentParts := 0; fi; Result := Result + n!/mul(op(j,cmpstn)!,j=1..NumberOfParts)*(NumberOfParts!/ mul(op(2,op(j,liste))!,j=1..NumberOfDifferentParts)); od; print(Result); end proc;
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LINKS
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Thomas Wieder, Home Page.
Thomas Wieder, (Old) Home Page.
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FORMULA
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For the number a(n) of such [L|R] configurations we have a(n) = n! sum_{i=1}^{n-1} delta2(i,n-i)/(i!*(n-i)!) where delta2(n,n-i) = 2 if i <> (n-i) and 1 if i = (n-i).
a(n)=A028399(n)-A126869(n), n>0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 07 2008]
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MAPLE
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A120672 := proc(n) local i, Term, Result; Result:=0; for i from 1 to n-1 do Term:=n!/(i!*(n-i)!); if i <> n-i then Term:=2*Term; fi; Result:=Result+Term; end do; print(Result); end proc;
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CROSSREFS
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Cf. A028399.
Sequence in context: A063599 A163479 A017293 this_sequence A108960 A111095 A073211
Adjacent sequences: A120669 A120670 A120671 this_sequence A120673 A120674 A120675
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KEYWORD
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nonn
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AUTHOR
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Thomas Wieder (thomas.wieder(AT)t-online.de), Jun 24 2006
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