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%I #6 Jun 13 2017 23:37:44
%S 1,1,1,1,2,1,1,3,3,1,1,5,7,4,1,1,9,17,13,5,1,1,19,45,43,21,6,1,1,47,
%T 135,153,89,31,7,1,1,137,463,603,401,161,43,8,1,1,465,1817,2657,1969,
%U 881,265,57,9,1,1,1819,8121,13111,10633,5191,1709,407,73,10,1
%N Triangle, read by rows, such that T(n,k) = T(n-1,k-1) + [T^2](n-2,k-1) with T(n,0) = T(n,n) = 1 for n>=0, k>=0.
%C Surprisingly, T(n,k) = [T^k](n-k,0) + [T^(k+1)](n-k-1,0), where T^k is the k-th power of T as a triangular matrix. See triangle A113993, where column k of A113993 equals column 0 of T^(k+1).
%F T(n, k) = [T^k](n-k, 0) + [T^(k+1)](n-k-1, 0), or, equivalently, T(n, k) = A113993(n-1, k-1) + A113993(n-1, k). From the definition, G.f. satisfies: A(x, y) = 1/(1-x) + x*y*A(x, y) + x^2*y*GF(T^2), where GF(T^2) equals the g.f. of the matrix square of T.
%e Triangle T begins:
%e 1;
%e 1,1;
%e 1,2,1;
%e 1,3,3,1;
%e 1,5,7,4,1;
%e 1,9,17,13,5,1;
%e 1,19,45,43,21,6,1;
%e 1,47,135,153,89,31,7,1;
%e 1,137,463,603,401,161,43,8,1;
%e 1,465,1817,2657,1969,881,265,57,9,1;
%e 1,1819,8121,13111,10633,5191,1709,407,73,10,1;
%e 1,8123,41075,72273,63297,33223,11759,3025,593,91,11,1; ...
%e Matrix square, T^2 (=A113987), begins:
%e 1;
%e 2,1;
%e 4,4,1;
%e 8,12,6,1;
%e 18,36,26,8,1;
%e 46,116,108,46,10,1;
%e 136,416,468,248,72,12,1; ...
%e where T(n,k) = T(n-1,k-1) + [T^2](n-2,k-1):
%e T(8,2) = 463 = T(7,1) + [T^2](6,1) = 47 + 416;
%e T(8,3) = 603 = T(7,2) + [T^2](6,2) = 135 + 468;
%e T(8,4) = 401 = T(7,3) + [T^2](6,3) = 153 + 248.
%o (PARI) T(n,k)=local(A,B);A=Mat(1);for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==1 || j==i,B[i,j]=1, B[i,j]=A[i-1,j-1]+(A^2)[i-2,j-1]);));A=B);A[n+1,k+1]
%Y Cf. A113984 (column 1), A113985 (column 2), A113986 (column 3), A113987 (column 4); A113988 (T^2), A113993.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Nov 12 2005
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