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%I #43 Jan 07 2024 16:31:29
%S 1,1,1,1,1,2,1,2,2,2,1,2,2,2,4,1,3,2,4,4,4,1,3,2,4,4,4,8,1,4,2,6,4,8,
%T 8,8,1,4,2,6,4,8,8,8,16,1,5,2,8,4,12,8,16,16,16,1,5,2,8,4,12,8,16,16,
%U 16,32,1,6,2,10,4,16,8,24,16,32,32,32,1,6,2,10,4,16,8,24,16,32,32,32,64
%N Triangle read by rows: T(n,k) is the number of paths from node 0 to k in a directed graph with n+1 vertices labeled 0, 1, ..., n and edges leading from i to i+1 for all i, and from i to i+2 for even i and from i to i-2 for odd i.
%H Andrew Howroyd, <a href="/A330617/b330617.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H E. Krom and M. M. Roughan, <a href="http://girlsangle.org/page/bulletin-archive/GABv13n03E.pdf">Path Counting and Eulerian Numbers</a>, Girls' Angle Bulletin, Vol. 13, No. 3 (2020), 8-10.
%F For k odd: T(n, k) = 2^((k-1)/2)*(ceiling(n/2) - (k-1)/2).
%F For k even: T(n, k) = 2^(k/2).
%F T(2*n-1, 2*k-1) = A130128(n, k).
%e First few rows of the triangle are:
%e 1;
%e 1, 1;
%e 1, 1, 2;
%e 1, 2, 2, 2;
%e 1, 2, 2, 2, 4;
%e 1, 3, 2, 4, 4, 4;
%e 1, 3, 2, 4, 4, 4, 8;
%e 1, 4, 2, 6, 4, 8, 8, 8;
%e 1, 4, 2, 6, 4, 8, 8, 8, 16;
%e 1, 5, 2, 8, 4, 12, 8, 16, 16, 16;
%e 1, 5, 2, 8, 4, 12, 8, 16, 16, 16, 32;
%e ...
%e For n=6 and k=3, T(6,3)=4 is the number of paths from node 0 to node 3 along the directed network: {0,1,2,3}, {0,2,3}, {0,2,4,5,3}, {0,1,2,4,5,3}.
%t Table[If[EvenQ@ k, 2^(k/2), 2^((k - 1)/2)*(Ceiling[n/2] - (k - 1)/2)], {n, 0, 12}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Mar 23 2020 *)
%o (PARI) T(n,k)={if(k%2, 2^(k\2)*((n+1)\2 - k\2), 2^(k/2))} \\ _Andrew Howroyd_, Mar 17 2020
%Y Cf. A130128.
%K nonn,easy,tabl,walk
%O 0,6
%A _Grace Work_, Mar 01 2020
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