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

0,5

Comments

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

Links

Table of n, a(n) for n=0..57.

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).

Example

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.

Prog

(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]

Crossrefs

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

Sequence in context: A204163 A122542 A227341 * A320019 A141343 A256678

Adjacent sequences: A098539 A098540 A098541 * A098543 A098544 A098545

Keyword

nonn,tabl

Author

Paul D. Hanna, Sep 16 2004

Status

approved