A102225 - OEIS

%I #6 Mar 30 2012 18:36:44

%S 1,1,1,2,-1,1,3,-5,3,1,7,-20,19,-5,1,17,-51,57,-33,11,1,75,-289,438,

%T -345,143,-21,1,346,-1426,2441,-2073,767,-97,43,1,4874,-22622,44289,

%U -46156,26231,-7713,1183,-85,1,49047,-259734,530214,-520395,272461,-103617,30735,1119,171,1,3009094,-15968025,35495592

%N Triangular matrix, read by rows, where row k is formed from the first differences of row (k-1) of its matrix square, with an appended '1' for the main diagonal.

%C Row sums are: {1,2,2,2,2,2,...}. Column 0 is A102226. Column 1 is A102227. Matrix square forms A102228.

%F T(n, k) = [T^2](n-1, k) - [T^2](n-1, k-1) for n>k>0, with T(n, n)=1 for n>=0 and T(n, 0) = [T^2](n-1, 0) for n>0.

%e Rows begin:

%e [1],

%e [1,1],

%e [2,-1,1],

%e [3,-5,3,1],

%e [7,-20,19,-5,1],

%e [17,-51,57,-33,11,1],

%e [75,-289,438,-345,143,-21,1],

%e [346,-1426,2441,-2073,767,-97,43,1],

%e [4874,-22622,44289,-46156,26231,-7713,1183,-85,1],...

%e The matrix square A102225^2 forms A102228:

%e [1],

%e [2,1],

%e [3,-2,1],

%e [7,-13,6,1],

%e [17,-34,23,-10,1],

%e [75,-214,224,-121,22,1],...

%e The first differences of the rows of A102228

%e form A102225 excluding the main diagonal of 1's.

%o (PARI) {T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,if(j==1,B[i,1]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]-(A^2)[i-1,j-1]));));A=B);return(A[n+1,k+1])}

%Y Cf. A102226, A102227, A102228.

%K sign,tabl

%O 0,4

%A _Paul D. Hanna_, Jan 01 2005