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A247629 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.

%I A247629 #5 Sep 23 2014 12:07:47
%S A247629 1,0,1,1,0,1,0,3,0,1,4,0,5,0,1,0,12,0,7,0,1,16,0,24,0,9,0,1,0,52,0,40,
%T A247629 0,11,0,1,68,0,116,0,60,0,13,0,1,0,236,0,216,0,84,0,15,0,1,304,0,568,
%U A247629 0,360,0,112,0,17,0,1,0,1108,0,1144,0,556,0,144
%N A247629 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.
%H A247629 Clark Kimberling, <a href="/A247629/b247629.txt">Table of n, a(n) for n = 0..1000</a>
%e A247629 First 9 rows:
%e A247629 1
%e A247629 0 ... 1
%e A247629 1 ... 0 ... 1
%e A247629 0 ... 3 ... 0 ... 1
%e A247629 4 ... 0 ... 5 ... 0 ... 1
%e A247629 0 ... 12 .. 0 ... 7 ... 0 ...1
%e A247629 16 .. 0 ... 24 .. 0 ... 9 ... 0 ... 1
%e A247629 0 ... 52 .. 0 ... 40 .. 0 ... 11 .. 0 ... 1
%e A247629 68 .. 0 ... 116 . 0 ... 60 .. 0 ... 13 .. 0 ... 1
%e A247629 T(4,2) counts these 5 paths given as vector sums applied to (0,0):
%e A247629 (1,1) + (1,1) + (1,1) + (1,-1)
%e A247629 (1,1) + (1,1) + (2,0)
%e A247629 (1,1) + (1,1) + (1,-1) + (1,1)
%e A247629 (1,1) + (2,0) + (1,1)
%e A247629 (1,1) + (1,-1) + (1,1) + (1,-1)
%t A247629 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] = If[n >= 2, t[n - 2, 0] + t[n - 1, 1], 0]; u = Table[t[n, k], {n, 0, 16}, {k, 0, n}]; TableForm[u] (* A247629 array *)
%t A247629 v = Flatten[u] (* A247629 sequence *)
%t A247629 Map[Total, u] (* A247630 *)
%Y A247629 Cf. A247623, A247629, A026300, A006319 (1st column of this triangle).
%K A247629 nonn,tabl,easy
%O A247629 0,8
%A A247629 _Clark Kimberling_, Sep 21 2014