A098542 Triangle T, read by rows, such that the matrix square shifts T left one column and up one row, with T(0,0)=T(1,0)=1 and T(n,0)=0 for n>1 and T(n,n)=1 for n>=0.

1, 1, 1, 0, 2, 1, 0, 2, 4, 1, 0, 2, 12, 8, 1, 0, 2, 44, 56, 16, 1, 0, 2, 236, 504, 240, 32, 1, 0, 2, 2028, 6776, 4720, 992, 64, 1, 0, 2, 29164, 146552, 139120, 40672, 4032, 128, 1, 0, 2, 719340, 5314680, 6583152, 2500832, 337344, 16256, 256, 1, 0, 2, 30943724

Offset: 0

Author

Paul D. Hanna, Sep 16 2004

Keywords

• nonn
• tabl

Comments

Column 2 forms A098543. Row sums form A098544. The absolute value of the matrix inverse equals A098539.

Examples

			Rows of T begin:
[1],
[1,1],
[0,2,1],
[0,2,4,1],
[0,2,12,8,1],
[0,2,44,56,16,1],
[0,2,236,504,240,32,1],
[0,2,2028,6776,4720,992,64,1],
[0,2,29164,146552,139120,40672,4032,128,1],
[0,2,719340,5314680,6583152,2500832,337344,16256,256,1],...
Rows of T^2 begin:
[1],
[2,1],
[2,4,1],
[2,12,8,1],
[2,44,56,16,1],
[2,236,504,240,32,1],...
showing that T shifts left and up under matrix square.
The matrix inverse of T begins:
[1],
[ -1,1],
[2,-2,1],
[ -6,6,-4,1],
[26,-26,20,-8,1],
[ -166,166,-140,72,-16,1],...
the absolute value of which equals triangle A098539.
		

Crossrefs

Cf. A098543, A098544, A098539 (absolute inverse).

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,j]=(A^0)[i-1,1],B[i,j]=(A^2)[i-1,j-1]));));A=B);A[n+1,k+1]

Formula

T(n+1, 1) = 2 for n>0; T(n+1, n) = 2^n, T(n+2, n) = 4^n - 2^n for n>=0. Matrix square: [T^2](n, k) = T(n+1, k+1). Matrix inverse: [T^-1](n, k) = (-1)^(n-k)*A098539(n, k). Matrix square inverse: [T^-2](n, k) = (-1)^(n-k)*A098539(n+1, k+1).