A098539 Lower triangular matrix T, read by rows, that shifts left one column under the matrix square of T, with T(n,0)=T(n,1) for n>0 and T(n,n)=1 for n>=0.

1, 1, 1, 2, 2, 1, 6, 6, 4, 1, 26, 26, 20, 8, 1, 166, 166, 140, 72, 16, 1, 1626, 1626, 1460, 888, 272, 32, 1, 25510, 25510, 23884, 16392, 6256, 1056, 64, 1, 664666, 664666, 639156, 479736, 215696, 46816, 4160, 128, 1, 29559718, 29559718, 28895052

Offset: 0

Author

Paul D. Hanna, Sep 13 2004

Keywords

• nonn
• tabl

Comments

Column 0 forms A002449, the number of different types of binary trees of height n. Row sums form A098540. Column 1 equals A098541. As a matrix, T satisfies [T^2](n,k) = T(n+1,k+1) for all n,k>=0, where [T^2] denotes the matrix square of T, with T(0,k)=[T^2](k,0) and T(k,k)=1 for all k>=0. This is a variant of triangle A078121.

Examples

			Rows of T begin:
[1],
[1,1],
[2,2,1],
[6,6,4,1],
[26,26,20,8,1],
[166,166,140,72,16,1],
[1626,1626,1460,888,272,32,1],
[25510,25510,23884,16392,6256,1056,64,1],
[664666,664666,639156,479736,215696,46816,4160,128,1],...
Matrix square T^2 begins:
[1],
[2,1],
[6,4,1],
[26,20,8,1],
[166,140,72,16,1],
[1626,1460,888,272,32,1],...
showing that T^2 is the same as T after shifting left one column.
		

Crossrefs

Cf. A002449, A098540, A078121.

Programs

PARI

T(n,k)=local(A,B,C,m);A=matrix(1,1);A[1,1]=1; for(m=2,n+1,B=A^2;C=matrix(m,m);for(i=1,m, for(j=1,i, if(i<3 || j==i || j>m-1,C[i,j]=1,if(j==1,C[i,j]=B[i-1,1],C[i,j]=B[i-1,j-1]));)); A=C);A[n+1,k+1]

Formula

T(n, 0) = A002449(n), T(n, n)=1 for n>=0; T(n, 1)=T(n, 0) for n>0.