A059294 2-boustrophedon transform applied to 1, 0, 0, 0, ...

1, 1, 1, 3, 13, 73, 505, 4143, 39313, 423401, 5101785, 67994611, 993048765, 15770916657, 270586214481, 4987678532991, 98297729816321, 2062591323728689, 45908909743929681, 1080350557160580163, 26800186367114537613, 698982753383076195897

Offset: 0

Author

N. J. A. Sloane, Jan 25 2001

Keywords

• nonn

Comments

The transform takes the original sequence s(0), s(1), s(2),.. and fills the 0th column of a lower triangular array with t(r,0) = s(r). The remaining columns of the triangular array are filled by a sum over the left neighbor plus two entries of the previous row: t(r,c) = t(r,c-1) +t(r-1,r-c) +t(r-1,r-c-1), 1<=c<=r. [The last term in this sum does not exist if r=c and is implicitly defined as t(r-1,-1)=0 then.] The result of the transform is the sequence of numbers t(r,r) along the diagonal of the triangular array for r>=0. - R. J. Mathar, Nov 12 2011

Programs

Maple

TBOUS := proc(a) local c,i,j,n: if whattype(a) <> list then RETURN([]); fi: n := min( nops(a), 60); for i from 0 to n-1 do c[i,0] := a[i+1]; od; for i from 1 to n-1 do for j from 1 to i-1 do c[i,j] := c[i,j-1] + c[i-1,i-j] + c[i-1,i-j-1]; od; c[i,i] := c[i,i-1] + c[i-1,0]; od; RETURN([seq(c[i,i],i=0..n-1)]); end: