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A120672 a(n) = 2 * A285917(n) for n >=2, a(0) = a(1) = 0.

%I A120672 #24 Apr 02 2024 08:19:47
%S A120672 0,0,2,12,22,60,104,252,438,1020,1792,4092,7264,16380,29332,65532,
%T A120672 118198,262140,475664,1048572,1912392,4194300,7683172,16777212,
%U A120672 30850272,67108860,123817124,268435452,496754308,1073741820,1992366124,4294967292,7988854198
%N A120672 a(n) = 2 * A285917(n) for n >=2, a(0) = a(1) = 0.
%C A120672 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.
%C A120672 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.
%F A120672 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).
%F A120672 a(n) = A028399(n) - A126869(n), n > 0. - _R. J. Mathar_, Aug 07 2008
%e A120672 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].
%p A120672 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;
%p A120672 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;
%t A120672 a[n_] := If[n == 0, 0, 2^(n+1) - 4 - Sum[Binomial[n, Quotient[k, 2]]* (-1)^(n-k), {k, 0, n}]];
%t A120672 Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Apr 02 2024, after _R. J. Mathar_'s formula *)
%Y A120672 Cf. A028399, A126869.
%K A120672 nonn
%O A120672 0,3
%A A120672 _Thomas Wieder_, Jun 24 2006
%E A120672 Simpler name referring to A285917 from _Joerg Arndt_, Jun 25 2019