login
Triangle T, read by rows, equal to the matrix square of triangle A113350, where T transforms column k of T into column k+1 of T.
8

%I #6 Jun 13 2017 22:51:25

%S 1,4,1,18,8,1,112,68,12,1,965,712,150,16,1,10957,9270,2184,264,20,1,

%T 156699,147174,37523,4912,410,24,1,2727793,2786270,754171,104476,9280,

%U 588,28,1,56306695,61662544,17502145,2531004,235025,15672,798,32,1

%N Triangle T, read by rows, equal to the matrix square of triangle A113350, where T transforms column k of T into column k+1 of T.

%C Also, T transforms column k of A113340^2 into column k+1 of A113340^2. Column 0: T(n,0) = A113356(n) = A113346(n+1) - 1, where A113346 equals column 0 of triangle A113345 (=A113340^2).

%F T(n, k) = sum_{j=0..n-k} T(n-k, j)*T(j+k-1, k-1) for n>=k>0 with T(n, 0) = A113346(n+1) - 1, for n>=0.

%e Triangle T begins:

%e 1;

%e 4,1;

%e 18,8,1;

%e 112,68,12,1;

%e 965,712,150,16,1;

%e 10957,9270,2184,264,20,1;

%e 156699,147174,37523,4912,410,24,1;

%e 2727793,2786270,754171,104476,9280,588,28,1;

%e 56306695,61662544,17502145,2531004,235025,15672,798,32,1; ...

%e where T transforms column k of T into column k+1:

%e at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];

%e at k=1, [Q^2]*[1,8,68,712,9270,...] = [1,12,150,2184,37523,...].

%o (PARI) T(n,k)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B); (matrix(#A,#A,r,c,if(r>=c,(A^(2*c))[r-c+1,1]))^2)[n+1,k+1]

%Y Cf. A113340, A113350, A113356 (column 0), A113357 (column 1), A113358 (column 2), A113359 (column 3); A091351.

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Nov 08 2005