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A247622 Triangular array: T(n,k) = number of paths from (0,0) to (n,k), each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis, and including no horizontal segment on the x-axis.

%I A247622 #6 Sep 23 2014 12:07:32
%S A247622 1,0,1,1,0,1,0,3,0,1,3,0,5,0,1,0,11,0,7,0,1,11,0,23,0,9,0,1,0,45,0,39,
%T A247622 0,11,0,1,45,0,107,0,59,0,13,0,1,0,197,0,205,0,83,0,15,0,1,197,0,509,
%U A247622 0,347,0,111,0,17,0,1,0,903,0,1061,0,541,0,143,0
%N A247622 Triangular array:  T(n,k) = number of paths from (0,0) to (n,k), each segment given by a vector (1,1), (1,-1), or (2,0), not crossing the x-axis, and including no horizontal segment on the x-axis.
%H A247622 Clark Kimberling, <a href="/A247622/b247622.txt">Table of n, a(n) for n = 0..1000</a>
%e A247622 First 9 rows:
%e A247622 1
%e A247622 0 ... 1
%e A247622 1 ... 0 ... 1
%e A247622 0 ... 3 ... 0 ... 1
%e A247622 3 ... 0 ... 5 ... 0 ... 1
%e A247622 0 ... 11 .. 0 ... 7 ... 0 ...1
%e A247622 11 .. 0 ... 23 .. 0 ... 9 ... 0 ... 1
%e A247622 0 ... 45 .. 0 ... 39 .. 0 ... 11 .. 0 ... 1
%e A247622 45 .. 0 ... 107 . 0 ... 59 .. 0 ... 13 .. 0 ... 1
%e A247622 T(3,1) counts these 3 paths given as vector sums applied to (0,0):
%e A247622 (1,1) + (1,-1), (2,0), (1,-1) + (1,1).
%t A247622 t[0, 0] = 1; t[1, 1] = 1; t[2, 0] = 1; t[2, 2] = 1; t[n_, k_] := t[n, k] = If[n >= 2 && k >= 1, t[n - 1, k - 1] + t[n - 1, k + 1] + t[n - 2, k], 0]; t[n_, 0] := t[n, 0] = t[n - 1, 1]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}];
%t A247622 v = Flatten[u] (* A247622 sequence *)
%t A247622 TableForm[u]   (* A247622 array *)
%t A247622 Map[Total, u]  (* A247623 *)
%Y A247622 Cf. A247623, A247629, A026300, A001003 (1st column of this triangle).
%K A247622 nonn,tabl,easy
%O A247622 0,8
%A A247622 _Clark Kimberling_, Sep 21 2014