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

%S 1,2,1,3,-2,1,7,-13,6,1,17,-34,23,-10,1,75,-214,224,-121,22,1,346,

%T -1080,1361,-712,55,-42,1,4874,-17748,26541,-19615,6616,-1097,86,1,

%U 49047,-210687,319527,-200868,71593,-32024,-1289,-170,1,3009094,-12958931,22536661,-19799672,9144014,-2280135,311880

%N Triangular matrix, read by rows, equal to the matrix square of A102225, such that the first differences of row k forms row (k+1) of A102225.

%C Column 0 is A102227 shift left. Column 1 is A102229.

%F T(n, k) = Sum_{j=0..k} A102225(n+1, j) for n>k>0, with T(n, n)=1 for n>=0 and T(n, 0) = A102226(n+1) for n>=0.

%e Rows begin:

%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 [346,-1080,1361,-712,55,-42,1],

%e [4874,-17748,26541,-19615,6616,-1097,86,1],...

%e Equals the matrix square of A102225, which starts:

%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 Each row k of A102228 equals the partial sums of

%e row (k+1) of A102225 (prior to main diagonal term).

%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^2)[n+1,k+1])}

%Y Cf. A102225, A102226, A102227, A102229.

%K sign,tabl

%O 0,2

%A _Paul D. Hanna_, Jan 01 2005