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A026765 a(n) = Sum_{k=0..n} T(n,k), T given by A026758.

%I A026765 #14 Nov 01 2019 03:54:57
%S A026765 1,2,4,9,18,41,82,188,376,867,1734,4020,8040,18735,37470,87735,175470,
%T A026765 412715,825430,1949624,3899248,9245721,18491442,44003717,88007434,
%U A026765 210121733,420243466,1006390014,2012780028,4833517551
%N A026765 a(n) = Sum_{k=0..n} T(n,k), T given by A026758.
%H A026765 G. C. Greubel, <a href="/A026765/b026765.txt">Table of n, a(n) for n = 0..1000</a>
%F A026765 Conjecture: G.f.: -(1-2*x-5*x^2+10*x^3 - sqrt(1-10*x^2+29*x^4-20*x^6) )/(2*x*(1-2*x-5*x^2+10*x^3)). -  Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
%F A026765 Conjecture: (n+1)*a(n) -2*a(n-1) +2*(-5*n+3)*a(n-2) +12*a(n-3) +(29*n-71)*a(n-4) -10*a(n-5) +20*(-n+5)*a(n-6)=0. - _R. J. Mathar_, Jun 30 2013
%F A026765 Conjecture: a(n) ~ (2+sqrt(5) + (-1)^n*(2-sqrt(5))) * 5^(n/2) / sqrt(2*Pi*n). - _Vaclav Kotesovec_, Feb 12 2014
%p A026765 T:= proc(n,k) option remember;
%p A026765    if n<0 then 0;
%p A026765    elif k=0 or k = n then 1;
%p A026765    elif type(n,'odd') and k <= (n-1)/2 then
%p A026765         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026765    else
%p A026765        procname(n-1,k-1)+procname(n-1,k) ;
%p A026765    end if ;
%p A026765 end proc;
%p A026765 seq(add(T(n,k), k=0..n), n=0..30); # _G. C. Greubel_, Oct 31 2019
%t A026765 T[n_, k_]:= T[n, k]= If[n<0, 0, 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,n}], {n, 0, 30}] (* _G. C. Greubel_, Oct 31 2019 *)
%o A026765 (Sage)
%o A026765 @CachedFunction
%o A026765 def T(n, k):
%o A026765     if (n<0): return 0
%o A026765     elif (k==0 or k==n): return 1
%o A026765     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 A026765     else: return T(n-1,k-1) + T(n-1,k)
%o A026765 [sum(T(n,k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Oct 31 2019
%Y A026765 Cf. A026758, A026759, A026760, A026761, A026762, A026763, A026764, A026766, A026767, A026768.
%K A026765 nonn
%O A026765 0,2
%A A026765 _Clark Kimberling_