[seqfan] Walks

[franktaw at netscape.net]

I just submitted a sequence https://oeis.org/draft/A187430 (Number of 
non-negative walks of length n with step sizes 1 and 2, starting and 
ending at 0.)

If one takes such walks without the restriction that it end at 0, then 
one can get T(n,k) to be the number of such walks ending at k. This 
starts:
1
0,1,1
2,1,1,2,1
2,5,6,3,3,3,1
11,11,13,17,13,7,6,4,1
24,41,52,44,43,40,25,14,10,5,1

The row sums of this sequence, which is the total number of walks of 
this sort of length n, starts
1,2,7,23,83,299,1107,4122,15523

This appears to be left-shifted A047002, array T(n,n) where T(n,k) = 
A047000(n,k) is the number of paths consisting of steps from (0,0) to 
(h,k) such that each step has length 1 directed up or right and no step 
touches the line y=x/2 unless x=0 or x=h.

This is not obvious to me. Can anyone verify it?

Franklin T. Adams-Watters