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A098539 Lower triangular matrix T, read by rows, that shifts left one column under the matrix square of T, with T(n,0)=T(n,1) for n>0 and T(n,n)=1 for n>=0.

%I A098539 #7 Jun 13 2017 22:13:27
%S A098539 1,1,1,2,2,1,6,6,4,1,26,26,20,8,1,166,166,140,72,16,1,1626,1626,1460,
%T A098539 888,272,32,1,25510,25510,23884,16392,6256,1056,64,1,664666,664666,
%U A098539 639156,479736,215696,46816,4160,128,1,29559718,29559718,28895052
%N A098539 Lower triangular matrix T, read by rows, that shifts left one column under the matrix square of T, with T(n,0)=T(n,1) for n>0 and T(n,n)=1 for n>=0.
%C A098539 Column 0 forms A002449, the number of different types of binary trees of height n. Row sums form A098540. Column 1 equals A098541. As a matrix, T satisfies [T^2](n,k) = T(n+1,k+1) for all n,k>=0, where [T^2] denotes the matrix square of T, with T(0,k)=[T^2](k,0) and T(k,k)=1 for all k>=0. This is a variant of triangle A078121.
%F A098539 T(n, 0) = A002449(n), T(n, n)=1 for n>=0; T(n, 1)=T(n, 0) for n>0.
%e A098539 Rows of T begin:
%e A098539 [1],
%e A098539 [1,1],
%e A098539 [2,2,1],
%e A098539 [6,6,4,1],
%e A098539 [26,26,20,8,1],
%e A098539 [166,166,140,72,16,1],
%e A098539 [1626,1626,1460,888,272,32,1],
%e A098539 [25510,25510,23884,16392,6256,1056,64,1],
%e A098539 [664666,664666,639156,479736,215696,46816,4160,128,1],...
%e A098539 Matrix square T^2 begins:
%e A098539 [1],
%e A098539 [2,1],
%e A098539 [6,4,1],
%e A098539 [26,20,8,1],
%e A098539 [166,140,72,16,1],
%e A098539 [1626,1460,888,272,32,1],...
%e A098539 showing that T^2 is the same as T after shifting left one column.
%o A098539 (PARI) T(n,k)=local(A,B,C,m);A=matrix(1,1);A[1,1]=1; for(m=2,n+1,B=A^2;C=matrix(m,m);for(i=1,m, for(j=1,i, if(i<3 || j==i || j>m-1,C[i,j]=1,if(j==1,C[i,j]=B[i-1,1],C[i,j]=B[i-1,j-1]));)); A=C);A[n+1,k+1]
%Y A098539 Cf. A002449, A098540, A078121.
%K A098539 nonn,tabl
%O A098539 0,4
%A A098539 _Paul D. Hanna_, Sep 13 2004