%I #8 Nov 01 2019 03:43:36
%S 1,6,31,156,784,3962,20173,103522,535294,2787700,14613710,77072816,
%T 408737760,2178631156,11666175215,62734622764,338660977020,
%U 1834690352066,9971834477972,54361287536706,297170702049966,1628670524735842
%N a(n) = T(2n,n-1), T given by A026769.
%H G. C. Greubel, <a href="/A026771/b026771.txt">Table of n, a(n) for n = 1..500</a>
%p T:= proc(n,k) option remember;
%p if n<0 then 0;
%p elif k=0 or k=n then 1;
%p elif n=2 and k=1 then 2;
%p elif k <= (n-1)/2 then
%p procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p else
%p procname(n-1,k-1)+procname(n-1,k) ;
%p end if ;
%p end proc;
%p seq(T(2*n, n-1), n=1..30); # _G. C. Greubel_, Nov 01 2019
%t T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]]]; Table[T[2*n, n-1], {n, 30}] (* _G. C. Greubel_, Nov 01 2019 *)
%o (Sage)
%o @CachedFunction
%o def T(n, k):
%o if (k==0 or k==n): return 1
%o elif (n==2 and k==1): return 2
%o elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
%o else: return T(n-1,k-1) + T(n-1,k)
%o [T(2*n, n-1) for n in (1..30)] # _G. C. Greubel_, Nov 01 2019
%Y Cf. A026769, A026770, A026772, A026773, A026774, A026775, A026776, A026777, A026778, A026779.
%K nonn
%O 1,2
%A _Clark Kimberling_