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A026790 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026780.

%I A026790 #12 Nov 03 2019 15:49:58
%S A026790 1,1,2,4,7,12,23,41,72,135,243,432,804,1455,2608,4836,8785,15838,
%T A026790 29306,53385,96654,178600,326019,592140,1093135,1998537,3638700,
%U A026790 6712659,12287071,22412784,41325279,75712253,138308808,254912873
%N A026790 a(n) = Sum_{k=0..floor(n/2)} T(n-k,k), T given by A026780.
%H A026790 G. C. Greubel, <a href="/A026790/b026790.txt">Table of n, a(n) for n = 0..1000</a>
%p A026790 T:= proc(n,k) option remember;
%p A026790     if n<0 then 0;
%p A026790     elif k=0 or k =n then 1;
%p A026790     elif k <= n/2 then
%p A026790         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026790     else
%p A026790         procname(n-1,k-1)+procname(n-1,k) ;
%p A026790     fi ;
%p A026790 end proc:
%p A026790 seq( add(T(n-k,k), k=0..floor(n/2)), n=0..40); # _G. C. Greubel_, Nov 02 2019
%t A026790 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 A026790 Table[Sum[T[n-k,k], {k, 0, Floor[n/2]}], {n,0,40}] (* _G. C. Greubel_, Nov 02 2019 *)
%o A026790 (Sage)
%o A026790 @CachedFunction
%o A026790 def T(n, k):
%o A026790     if (n<0): return 0
%o A026790     elif (k==0 or k==n): return 1
%o A026790     elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
%o A026790     else: return T(n-1,k-1) + T(n-1,k)
%o A026790 [sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..40)] # _G. C. Greubel_, Nov 02 2019
%Y A026790 Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026787, A026788, A026789.
%K A026790 nonn
%O A026790 0,3
%A A026790 _Clark Kimberling_