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A113355 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.

%I A113355 #6 Jun 13 2017 22:51:25
%S A113355 1,4,1,18,8,1,112,68,12,1,965,712,150,16,1,10957,9270,2184,264,20,1,
%T A113355 156699,147174,37523,4912,410,24,1,2727793,2786270,754171,104476,9280,
%U A113355 588,28,1,56306695,61662544,17502145,2531004,235025,15672,798,32,1
%N A113355 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 A113355 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 A113355 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 A113355 Triangle T begins:
%e A113355 1;
%e A113355 4,1;
%e A113355 18,8,1;
%e A113355 112,68,12,1;
%e A113355 965,712,150,16,1;
%e A113355 10957,9270,2184,264,20,1;
%e A113355 156699,147174,37523,4912,410,24,1;
%e A113355 2727793,2786270,754171,104476,9280,588,28,1;
%e A113355 56306695,61662544,17502145,2531004,235025,15672,798,32,1; ...
%e A113355 where T transforms column k of T into column k+1:
%e A113355 at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];
%e A113355 at k=1, [Q^2]*[1,8,68,712,9270,...] = [1,12,150,2184,37523,...].
%o A113355 (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 A113355 Cf. A113340, A113350, A113356 (column 0), A113357 (column 1), A113358 (column 2), A113359 (column 3); A091351.
%K A113355 nonn,tabl
%O A113355 0,2
%A A113355 _Paul D. Hanna_, Nov 08 2005