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A026779 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026769.

%I A026779 #8 Nov 01 2019 22:23:41
%S A026779 1,1,2,3,6,10,17,32,56,97,181,322,567,1053,1892,3369,6241,11286,20255,
%T A026779 37463,68044,122809,226896,413376,749159,1382990,2525162,4590351,
%U A026779 8468738,15487526,28218889,52035094,95273724,173898941
%N A026779 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026769.
%H A026779 G. C. Greubel, <a href="/A026779/b026779.txt">Table of n, a(n) for n = 0..1000</a>
%p A026779 T:= proc(n,k) option remember;
%p A026779    if n<0 then 0;
%p A026779    elif k=0 or k=n then 1;
%p A026779    elif n=2 and k=1 then 2;
%p A026779    elif k <= (n-1)/2 then
%p A026779         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026779    else
%p A026779        procname(n-1,k-1)+procname(n-1,k) ;
%p A026779    end if ;
%p A026779 end proc;
%p A026779 seq(add(T(n-k,k), k=0..floor(n/2)), n=0..30); # _G. C. Greubel_, Nov 01 2019
%t A026779 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[Sum[T[n-k,k], {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, Nov 01 2019 *)
%o A026779 (Sage)
%o A026779 @CachedFunction
%o A026779 def T(n, k):
%o A026779     if (k==0 or k==n): return 1
%o A026779     elif (n==2 and k==1): return 2
%o A026779     elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
%o A026779     else: return T(n-1,k-1) + T(n-1,k)
%o A026779 [sum(T(n-k,k) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Nov 01 2019
%Y A026779 Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026778.
%K A026779 nonn
%O A026779 0,3
%A A026779 _Clark Kimberling_