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.

1, 4, 1, 18, 8, 1, 112, 68, 12, 1, 965, 712, 150, 16, 1, 10957, 9270, 2184, 264, 20, 1, 156699, 147174, 37523, 4912, 410, 24, 1, 2727793, 2786270, 754171, 104476, 9280, 588, 28, 1, 56306695, 61662544, 17502145, 2531004, 235025, 15672, 798, 32, 1

Offset: 0

Author

Paul D. Hanna, Nov 08 2005

Keywords

• nonn
• tabl

Comments

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).

Examples

			Triangle T begins:
1;
4,1;
18,8,1;
112,68,12,1;
965,712,150,16,1;
10957,9270,2184,264,20,1;
156699,147174,37523,4912,410,24,1;
2727793,2786270,754171,104476,9280,588,28,1;
56306695,61662544,17502145,2531004,235025,15672,798,32,1; ...
where T transforms column k of T into column k+1:
at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];
at k=1, [Q^2]*[1,8,68,712,9270,...] = [1,12,150,2184,37523,...].
		

Crossrefs

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

Programs

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]

Formula

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.