Lozanic's triangle A034851 -- Matrix Log Holds Key

[Alonso Del Arte]

How exactly does the matrix log hold the key? (I don't know what amatrix log is, it might be one of those things I was supposed to learnin high school but didn't).
Something that might be obvious to everyone else, but which is alittle bit of a revelation to me, is that Lozanić's triangle can bederived from Pascal's triangle. If we inflate Pascal's triangle thus
00 00 1 00 0 0 00 1 0 1 00 0 0 0 0 00 1 0 2 0 1 00 0 0 0 0 0 0 00 1 0 3 0 3 0 1 00 0 0 0 0 0 0 0 0 00 1 0 4 0 6 0 4 0 1 0...
and subtract it from the original triangle (taking care to propagatethe changes to the rows below as necessary) the result is Lozanić'striangle.
Alonso





On 3/4/06, Paul D. Hanna <pauldhanna at juno.com> wrote:>> Christian (and Seqfans),>         Using the trend observed in the matrix log, I calculated more rows> of your triangle.> If your original triangle continues as:> 1;> 1,1;> 1,0,1;> 1,1,1,1;> 1,0,2,0,1;> 1,1,2,2,1,1;> 1,0,3,0,3,0,1;> 1,1,3,3,3,3,1,1;> 1,0,4,0,6,0,4,0,1;> 1,1,4,4,6,6,4,4,1,1;> 1,0,5,0,10,0,10,0,5,0,1;> 1,1,5,5,10,10,10,10,5,5,1,1; ...>> then this triangle is already in the OEIS as A051159 ...> is that correct?>> At any rate, the inverse is not in the OEIS, and is worthy.>> Also, it may be interesting to note the following related sequences.>> Antidiagonal sums = A053602> a(n)=a(n-1)-(-1)^n*a(n-2), a(0)=0, a(1)=1.> 1,1,2,1,3,2,5,3,8,5,13,8,21,13,34,21,55,34,89,55,144,>> Row squared sums = A063886> Number of n-step walks on a line starting from the origin but not returning> to it.> 1,2,2,4,6,12,20,40,70,140,252,504,924,1848,3432,6864,>> Paul>> On Sat, 4 Mar 2006 00:29:57 -0500 "Paul D. Hanna" <pauldhanna at jun!
o.com>> writes:> > Dear Christian (and Seqfans)> >          Very interesting triangle ... when one takes the matrix log,> > the pattern is obvious and significant.> > This pattern accounts for the unusual sign pattern in the matrix> > inverse:> > log(T) => > 0;> > 1, 0;> > 1, 0, 0;> > 0, 1, 1, 0;> > 0, 0, 2, 0, 0;> > 0, 0, 0, 2, 1, 0;> > 0, 0, 0, 0, 3, 0, 0;> > 0, 0, 0, 0, 0, 3, 1, 0; ...> >> > This pattern also says something about the e.g.f of your original> > triangle.> >> > Paul