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A247311 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1).

%I A247311 #14 Nov 10 2024 21:45:46
%S A247311 1,0,0,1,1,0,2,2,1,4,5,3,9,12,8,21,29,20,50,70,49,120,169,119,289,408,
%T A247311 288,697,985,696,1682,2378,1681,4060,5741,4059,9801,13860,9800,23661,
%U A247311 33461,23660,57122,80782,57121,137904,195025,137903,332929,470832,332928
%N A247311 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1).
%C A247311 Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, and for 0 < i <= n, s(i) is in {0,1,2}, and s(i) - s(i-1) is in {-1,0,1}.
%C A247311 (row 0, the bottom row): A024537;
%C A247311 (row 1, the middle row): A000129;
%C A247311 (row 2, the top row): A048739;
%C A247311 (n-th column sum): A000129.
%H A247311 Clark Kimberling, <a href="/A247311/b247311.txt">Table of n, a(n) for n = 0..1000</a>
%e A247311 First 10 columns:
%e A247311   0 .. 0 .. 1 .. 3 .. 8 ... 20 .. 49 .. 119 .. 288 .. 696
%e A247311   0 .. 1 .. 2 .. 5 .. 12 .. 29 .. 70 .. 169 .. 408 .. 985
%e A247311   1 .. 1 .. 2 .. 4 .. 9 ... 21 .. 50 .. 120 .. 289 .. 697
%e A247311 T(3,2) counts these 3 paths, given as vector sums applied to (0,0):
%e A247311   (1,1) + (1,1) + (1,0); (1,1) + (1,0) + (1,1); (1,0) + (1,1) + (1,1).
%t A247311 t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 2] = 0;
%t A247311 t[n_, 0] := t[n, 0] = t[n - 1, 0] + t[n - 1, 1];
%t A247311 t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2];
%t A247311 t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 1, 2]
%t A247311 TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
%t A247311 Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* A247311 *)
%Y A247311 Cf.  A247049, A247309, A247310, A000129, A024537, A048739.
%K A247311 nonn,tabf,easy
%O A247311 0,7
%A A247311 _Clark Kimberling_, Sep 12 2014