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%I
%S 0,1,1,1,2,1,3,1,5,2,3,1,11,1,3,3,15,1,11,1,11,3,3,1,45,2,3,5,11,1,19,
%T 1,52,3,3,3,62,1,3,3,45,1,19,1,11,11,3,1,200,2,11,3,11,1,45,3,45,3,3,
%U 1,113,1,3,11,203,3,19,1,11,3,19,1,355,1,3,11,11,3,19,1,200,15,3,1,113,3
%N Number of complementing systems of subsets of {0, 1, ..., n-1}.
%C Number of collections {S_1, S_2, ..., S_k} of subsets of {0, 1, ..., n-1}, each subset containing 0, such that every element x of {0,1, ..., n-1} can be uniquely expressed as x=x_1+x_2+ ...+ x_k with x_i in S_i for all i=1..k.
%D C. T. Long, Addition Theorems for sets of Integers, Pacific J. Math. 23 (1967), 107-112.
%D A. O. Munagi, k-Complementing Subsets of Nonnegative Integers, IJMMS 2005:2, (2005), 215-224.
%H N. J. A. Sloane, <a href="/A104725/b104725.txt">Table of n, a(n) for n = 0..10000</a>
%H A. O. Munagi, <a href="/A104725/a104725.txt">Notes on A104725</a>
%H A. O. Munagi, <a href="http://www.hindawi.com/GetArticle.aspx?doi=10.1155/IJMMS.2005.215">k-Complementing Subsets of Nonnegative Integers</a>, IJMMS 2005:2 (2005), 215-224.
%F a(0)=0, a(1)=1, a(n)=Sum(ordfac(n,k)*Bell(k-1),k=1..Omega(n)), n>1, where ordfac(n,k)=number of ordered factorizations of n into k factors.
%F a(n)= A074206(n) if A001222(n)=1, 2.
%e a(6) = 3: {{0,1,2,3,4,5}}, {{0,1,2},{0,3}} and {{0,1},{0,2,4}}.
%e Thus since {{0,1,2},{0,3}} is a complementing system of subsets of {0,1,2,3,4,5} we have 0=0+0, 1=1+0, 2=2+0, 3=0+3, 4=1+3, 5=2+3.
%p a:=proc(n::integer) local u,r,i,j,k; if n<1 then return 0; elif n=1 then return 1; end if; u:=map(x->x[2],ifactors(n)[2]); r:=add(u[i],i=1..nops(u)); add(add((-1)^i*binomial(k,i)*product(binomial(u[j]+k-i-1,u[j]),j=1..nops(u)),i=0..k-1)*bell(k-1),k=1..r); end proc: seq(a(n),n=0..90);
%t nmax=85; a[n_] := (u = FactorInteger[n][[All, 2]]; r = Total[u]; Sum[ Sum[(-1)^i*Binomial[k, i]* Product[ Binomial[ u[[j]]+k-i-1, u[[j]] ], {j, 1, Length[u]}], {i, 0, k-1}]*BellB[k-1], {k, 1, r}]); a[0] = 0; a[1] = 1; Table[a[n], {n, 0, nmax}](* From Jean-François Alcover, Nov 18 2011, after Maple *)
%Y Cf. A074206, A002033.
%K nonn,nice,core
%O 0,5
%A A. O. Munagi (amunagi(AT)yahoo.com), Mar 20, 2005; Dec 20, 2006
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