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A026733 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026725.

%I A026733 #10 Oct 26 2019 15:20:39
%S A026733 1,1,3,5,13,23,57,103,249,455,1083,1993,4693,8679,20275,37633,87377,
%T A026733 162643,375789,701075,1613413,3015563,6916957,12948083,29617161,
%U A026733 55513327,126678893,237705547,541325021,1016736115,2311294377
%N A026733 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026725.
%H A026733 G. C. Greubel, <a href="/A026733/b026733.txt">Table of n, a(n) for n = 0..1000</a>
%F A026733 Conjecture: (-n+2)*a(n) +(n-2)*a(n-1) +2*(4*n-13)*a(n-2) +8*(-n+4)*a(n-3) +5*(-3*n+14)*a(n-4) +(15*n-94)*a(n-5) +2*(-2*n+9)*a(n-6) +4*(n-6)*a(n-7)=0. - _R. J. Mathar_, Oct 26 2019
%p A026733 A026733 := proc(n)
%p A026733     add(A026725(n,k),k=0..floor(n/2)) ;
%p A026733 end proc:
%p A026733 seq(A026733(n),n=0..10) ; # _R. J. Mathar_, Oct 26 2019
%t A026733 T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[OddQ[n] && 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, 0, Floor[n/2]}], {n, 0, 30}] (* _G. C. Greubel_, Oct 26 2019 *)
%o A026733 (PARI) T(n,k) = if(k==n || k==0, 1, if(2*k==n-1, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
%o A026733 vector(31, n, sum(k=0,floor(n-1/2), T(n-1,k)) ) \\ _G. C. Greubel_, Oct 26 2019
%o A026733 (Sage)
%o A026733 @CachedFunction
%o A026733 def T(n, k):
%o A026733     if (k==0 or k==n): return 1
%o A026733     elif (mod(n,2)==1 and k==(n-1)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
%o A026733     else: return T(n-1, k-1) + T(n-1, k)
%o A026733 [sum(T(n, k) for k in (0..floor(n/2))) for n in (0..30)] # _G. C. Greubel_, Oct 26 2019
%K A026733 nonn
%O A026733 0,3
%A A026733 _Clark Kimberling_