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%I #7 Sep 23 2014 16:42:27
%S 1,1,2,4,10,20,50,104,258,552,1362,2972,7306,16172,39650,88720,217090,
%T 489872,1196834,2719028,6634890,15157188,36949266,84799992,206549250,
%U 475894200,1158337650,2677788492,6513914634,15102309468,36718533570,85347160608
%N Number of paths from (0,0) to the line x = n, 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.
%C a(n) = sum of the numbers in row n of the triangle at A247629.
%H Clark Kimberling, <a href="/A247630/b247630.txt">Table of n, a(n) for n = 0..1000</a>
%F Conjecture: -(n+1)*(n-2)*a(n) -(n-1)*(n-4)*a(n-1) +2*(3*n-2)*(n-2)*a(n-2) +2*(3*n-5)*(n-3)*a(n-3) +(-n^2+7*n-2)*a(n-4) -(n-1)*(n-6)*a(n-5)=0. - _R. J. Mathar_, Sep 23 2014
%e First 9 rows:
%e 1
%e 0 ... 1
%e 1 ... 0 ... 1
%e 0 ... 3 ... 0 ... 1
%e 4 ... 0 ... 5 ... 0 ... 1
%e 0 ... 12 .. 0 ... 7 ... 0 ...1
%e 16 .. 0 ... 24 .. 0 ... 9 ... 0 ... 1
%e 0 ... 52 .. 0 ... 40 .. 0 ... 11 .. 0 ... 1
%e 68 .. 0 ... 116 . 0 ... 60 .. 0 ... 13 .. 0 ... 1
%e T(4,2) counts these 5 paths given as vector sums applied to (0,0):
%e (1,1) + (1,1) + (1,1) + (1,-1)
%e (1,1) + (1,1) + (2,0)
%e (1,1) + (1,1) + (1,-1) + (1,1)
%e (1,1) + (2,0) + (1,1)
%e (1,1) + (1,-1) + (1,1) + (1,-1)
%e a(4) = 4 + 0 + 5 + 0 + 1 = 10.
%t 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 v = Flatten[u] (* A247629 sequence *)
%t Map[Total, u] (* A247630 *)
%Y Cf. A247629, A247623.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Sep 21 2014
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