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%I #6 Jun 13 2017 22:10:13
%S 1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,5,5,2,1,1,1,8,10,5,2,1,1,1,15,22,14,
%T 5,2,1,1,1,28,47,34,14,5,2,1,1,1,61,113,88,42,14,5,2,1,1,1,133,269,
%U 223,116,42,14,5,2,1,1,1,328,705,609,333,132,42,14,5,2,1,1,1,807,1843,1660
%N Triangle T, read by rows, such that the matrix square shifts T one place diagonally left and upward, with T(n,0)=T(n,n)=1 for n>=0.
%C Column with index 1 forms the row sums shift right. The convergent of the rows in reverse order is the Catalan sequence (A000108).
%F T(n, k) = Sum_{i=0..n-1} T(n-2, i)*T(i, k-1) for n>1 and k>0; T(n, 0)=T(n, n)=1.
%e Rows begin:
%e [1],
%e [1,1],
%e [1,1,1],
%e [1,2,1,1],
%e [1,3,2,1,1],
%e [1,5,5,2,1,1],
%e [1,8,10,5,2,1,1],
%e [1,15,22,14,5,2,1,1],
%e [1,28,47,34,14,5,2,1,1],
%e [1,61,113,88,42,14,5,2,1,1],
%e [1,133,269,223,116,42,14,5,2,1,1],...
%e The matrix square of T is given by:
%e [1],
%e [2,1],
%e [3,2,1],
%e [5,5,2,1],
%e [8,10,5,2,1],
%e [15,22,14,5,2,1],
%e [28,47,34,14,5,2,1],
%e [61,113,88,42,14,5,2,1],
%e [133,269,223,116,42,14,5,2,1],...
%e which equals T shift one place diagonally left and upward.
%o (PARI) T(n,k)=if(n<k || k<0,0,if(n==k || k==0,1,sum(i=0,n-1,T(n-2,i)*T(i,k-1));))
%Y Cf. A096592, A000108.
%K nonn,tabl
%O 0,8
%A _Paul D. Hanna_, Jun 28 2004
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