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A120672
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a(n) = 2 * A285917(n) for n >=2, a(0) = a(1) = 0.
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1
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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, 16777212, 30850272, 67108860, 123817124, 268435452, 496754308, 1073741820, 1992366124, 4294967292, 7988854198
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OFFSET
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0,3
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COMMENTS
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Previous name was: 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.
A028399(n) = 2*2^n - 4 with n=1,2,3,... is an upper limit for a(n) because Sum_{i=1..n-1} 2*n!/(i!*(n-i)!) = 2*2^n - 4. a(n) follows from all distinct ordered 2-tuples of positive integers whose elements sum to n. See the first Maple program below.
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LINKS
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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).
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EXAMPLE
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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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MAPLE
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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;
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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MATHEMATICA
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a[n_] := If[n == 0, 0, 2^(n+1) - 4 - Sum[Binomial[n, Quotient[k, 2]]* (-1)^(n-k), {k, 0, n}]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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