%I
%S 1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,2,3,2,1,1,1,2,3,4,2,1,1,1,2,3,5,4,
%T 2,1,1,1,2,3,5,7,4,2,1,1,1,2,3,5,8,8,4,2,1,1,1,2,3,5,8,12,8,4,2,1,1,1,
%U 2,3,5,8,13,15,8,4,2,1,1,1,2,3,5,8,13,20,16
%N A pathcounting array, read by rows: T(i,j)=number of paths from (0,0) to (ij,j) using steps (1 unit right and 1 unit up) or (1 unit right and 2 units up).
%C If m >= 1 and n >= 2, then T(m+n1,m) is the number of strings (s(1),s(2),...,s(n)) of nonnegative integers satisfying s(n)=m and 1<=s(k)s(k1)<=2 for k=2,3,...,n.
%H C. Kimberling, <a href="https://www.fq.math.ca/Scanned/404/kimberling.pdf">Pathcounting and Fibonacci numbers</a>, Fib. Quart. 40 (4) (2002) 328338, Example 1D.
%F T(i, 0)=T(i, i)=1 for i >= 0; T(i, 1)=1 for i >= 1; T(i, j)=T(i2, j1)+T(i3, j2) for 2<=j<=i1, i >= 3.
%e 7=T(8,5) counts these strings: 0135, 0235, 0245, 1235, 1245, 1345, 2345.
%e Rows: {1}; {1,1}; {1,1,1}; {1,1,2,1}; {1,1,2,2,1}; ...
%Y T(2n, n)=A000045(n+1), the Fibonacci numbers.
%K nonn,tabl,walk
%O 1,9
%A _Clark Kimberling_, May 07 2000
