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%I #4 Apr 25 2016 12:09:15
%S 1,1,1,1,1,1,-1,-2,-3,1,-1,1,3,-3,1,-2,-3,-3,2,-3,1,-1,2,2,0,2,-3,1,
%T -2,-4,-3,0,-1,2,-3,1,-2,2,3,-3,3,-1,2,-3,1,-2,-4,-4,3,-4,2,-1,2,-3,1,
%U -1,3,4,-1,0,-1,2,-1,2,-3,1,-2,-5,-5,1,1,-1,-2,2,-1,2,-3,1,-2,3,4,-4,3,-1,2
%N Triangle T(n,k) read by rows. n>3,k=1 T(n,k)=A002321(n-1). The rest of the table is described by the recurrence in the Excel formula.
%C Matrix inverse of A180051. Where the Excel formula says "randbetween(-9;9)" this table has the values of the Mertens function in the first column. Help with translating the spreadsheet formula would be appreciated.
%F Contribution from _Mats Granvik_, Aug 11 2010: (Start)
%F [from Wouter Meeussen, seqfan]
%F a(r,c)=0 /; c>r
%F a(r,c)=1 /; r<=3
%F a(r,1)=sum(Amu(k),k=1..r)
%F a(r,c)=a(r,c-1)-sum(a(r-j,c), j=1..c-1)/; c<=3
%F a(r,c)=sum(a(r-j,c-1), j=1..c-2)-sum(a(r-j,c), j=1..c-1)
%F (End)
%e Table begins:
%e 1,
%e 1,1,
%e 1,1,1,
%e -1,-2,-3,1,
%e -1,1,3,-3,1,
%e -2,-3,-3,2,-3,1,
%e -1,2,2,0,2,-3,1,
%e -2,-4,-3,0,-1,2,-3,1,
%e -2,2,3,-3,3,-1,2,-3,1,
%e -2,-4,-4,3,-4,2,-1,2,-3,1,
%e -1,3,4,-1,0,-1,2,-1,2,-3,1,
%e -2,-5,-5,1,1,-1,-2,2,-1,2,-3,1,
%e -2,3,4,-4,3,-1,2,-2,2,-1,2,-3,1,
%e -3,-6,-5,3,-4,2,-2,1,-2,2,-1,2,-3,1,
%t Contribution from _Mats Granvik_, Aug 11 2010: (Start)
%t [from Wouter Meeussen, seqfan]
%t Clear[a];
%t a[r_,c_]:=0 /; c>r;
%t a[r_,c_]:=1 /; r<=3;
%t a[r_,1]:=Sum[MoebiusMu[k],{k,0,r-1}];
%t a[r_,c_]:=a[r,c-1]-Sum[a[r-j,c], {j,1,c-1}]/; c<=3;
%t a[r_,c_]:=a[r,c]=Sum[a[r-j,c-1], {j,1,c-2}]-Sum[a[r-j,c], {j,1,c-1}];
%t (m=Table[a[i,j],{i,14},{j,14}])//ColumnForm
%t (End)
%o (Excel) Using European dot comma style:
%o =if(row()>=column();if(row()<=3;1;if(column()=1; randbetween(-9;9);if(or(column()=2;column()=3);sum(indirect(address(row();column()-1; 4)))-sum(indirect(address(row()-column()+1; column(); 4)&":"&address(row()-1; column(); 4); 4));sum(indirect(address(row()-column()+2; column()-1; 4)&":"&address(row()-1; column()-1; 4); 4))-sum(indirect(address(row()-column()+1; column(); 4)&":"&address(row()-1; column(); 4); 4)))));0)
%Y Cf. A002321, A180051.
%K sign,tabl
%O 1,8
%A _Mats Granvik_, Aug 08 2010
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