A180050 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.

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, -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, -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

Offset: 1

Author

Mats Granvik, Aug 08 2010

Keywords

• sign
• tabl

Comments

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.

Examples

			Table begins:
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,
-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,
-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,-2,2,-1,2,-3,1,
-3,-6,-5,3,-4,2,-2,1,-2,2,-1,2,-3,1,
		

Crossrefs

Cf. A002321, A180051.

Programs

Excel

Using European dot comma style:
=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)

Mathematica

Contribution from Mats Granvik, Aug 11 2010: (Start)
[from Wouter Meeussen, seqfan]
Clear[a];
a[r_,c_]:=0 /; c>r;
a[r_,c_]:=1 /; r<=3;
a[r_,1]:=Sum[MoebiusMu[k],{k,0,r-1}];
a[r_,c_]:=a[r,c-1]-Sum[a[r-j,c], {j,1,c-1}]/; c<=3;
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}];
(m=Table[a[i,j],{i,14},{j,14}])//ColumnForm
(End)

Formula

Contribution from Mats Granvik, Aug 11 2010: (Start)

[from Wouter Meeussen, seqfan]

a(r,c)=0 /; c>r

a(r,c)=1 /; r<=3

a(r,1)=sum(Amu(k),k=1..r)

a(r,c)=a(r,c-1)-sum(a(r-j,c), j=1..c-1)/; c<=3

a(r,c)=sum(a(r-j,c-1), j=1..c-2)-sum(a(r-j,c), j=1..c-1)

(End)