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A026788 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026780.

%I A026788 #8 Nov 03 2019 02:09:37
%S A026788 1,1,4,6,20,34,105,191,563,1071,3057,6007,16745,33729,92332,189662,
%T A026788 511812,1068178,2849404,6025594,15921514,34043204,89242582,192621212,
%U A026788 501574732,1091400122,2825710822,6192005260,15952433940,35172854946
%N A026788 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026780.
%H A026788 G. C. Greubel, <a href="/A026788/b026788.txt">Table of n, a(n) for n = 0..1000</a>
%p A026788 T:= proc(n,k) option remember;
%p A026788     if n<0 then 0;
%p A026788     elif k=0 or k =n then 1;
%p A026788     elif k <= n/2 then
%p A026788         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026788     else
%p A026788         procname(n-1,k-1)+procname(n-1,k) ;
%p A026788     fi ;
%p A026788 end proc:
%p A026788 seq( add(T(n,k), k=0..floor(n/2)), n=0..30); # _G. C. Greubel_, Nov 02 2019
%t A026788 T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
%t A026788 Table[Sum[T[n, k], {k, 0, Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, Nov 02 2019 *)
%o A026788 (Sage)
%o A026788 @CachedFunction
%o A026788 def T(n, k):
%o A026788     if (n<0): return 0
%o A026788     elif (k==0 or k==n): return 1
%o A026788     elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
%o A026788     else: return T(n-1,k-1) + T(n-1,k)
%o A026788 [sum(T(n,k) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Nov 02 2019
%Y A026788 Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026787, A026789, A026790.
%K A026788 nonn
%O A026788 0,3
%A A026788 _Clark Kimberling_