A026681 - OEIS

%I #15 Jun 04 2020 06:36:45

%S 1,1,1,1,3,1,1,5,5,1,1,7,10,7,1,1,9,17,17,9,1,1,11,26,44,26,11,1,1,13,

%T 37,87,87,37,13,1,1,15,50,150,174,150,50,15,1,1,17,65,237,324,324,237,

%U 65,17,1,1,19,82,352,561,822,561,352,82,19,1

%N Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-1,k) if k or n-k is of form 2h for h=1,2,...,[ n/4 ], else T(n,k)=T(n-1,k-1)+T(n-2,k-1)+T(n-1,k).

%H Robert Israel, <a href="/A026681/b026681.txt">Table of n, a(n) for n = 1..10011</a>

%F T(n, k) is the number of paths from (0, 0) to (n-k, k) in the directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, j)-to-(i+1, j+1) for i even and j >= i and for j even and i >= j.

%p T:= proc(n,k) option remember;

%p if k=0 or k=n then return 1 fi;

%p if min(k,n-k)::even then procname(n-1,k-1)+procname(n-1,k)

%p else procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k)

%p fi

%p end proc:

%p seq(seq(T(n,k),k=0..n),n=0..15); # _Robert Israel_, Jul 16 2019

%t T[n_, k_] := T[n, k] = With[{}, If[k == 0 || k == n, Return[1]]; If[EvenQ[ Min[k, n-k]], T[n-1, k-1] + T[n-1, k], T[n-1, k-1] + T[n-2, k-1] + T[n-1, k]]];

%t Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 04 2020, after Maple *)

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_