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A026767 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026758.

%I A026767 #12 Nov 01 2019 03:54:47
%S A026767 1,3,7,16,34,75,157,345,721,1588,3322,7342,15382,34117,71587,159322,
%T A026767 334792,747507,1572937,3522561,7421809,16667530,35158972,79162689,
%U A026767 167170123,377291856,797535322,1803925336,3816705364
%N A026767 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026758.
%C A026767 Partial sums of A026765.
%H A026767 G. C. Greubel, <a href="/A026767/b026767.txt">Table of n, a(n) for n = 0..1000</a>
%F A026767 Conjecture: (n+1)*a(n) +(-n-3)*a(n-1) +2*(-5*n+4)*a(n-2) +2*(5*n+3)*a(n-3) +(29*n-83)*a(n-4) +(-29*n+61)*a(n-5) +10*(-2*n+11)*a(n-6) +20*(n-5)*a(n-7)=0. - _R. J. Mathar_, Jun 30 2013
%p A026767 T:= proc(n,k) option remember;
%p A026767    if n<0 then 0;
%p A026767    elif k=0 or k = n then 1;
%p A026767    elif type(n,'odd') and k <= (n-1)/2 then
%p A026767         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026767    else
%p A026767        procname(n-1,k-1)+procname(n-1,k) ;
%p A026767    end if ;
%p A026767 end proc;
%p A026767 seq( add(add(T(j,k), k=0..n), j=0..n), n=0..30); # _G. C. Greubel_, Oct 31 2019
%t A026767 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[j,k], {k,0,n}, {j,0,n}], {n, 0, 30}] (* _G. C. Greubel_, Oct 31 2019 *)
%o A026767 (Sage)
%o A026767 @CachedFunction
%o A026767 def T(n, k):
%o A026767     if (n<0): return 0
%o A026767     elif (k==0 or k==n): return 1
%o A026767     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 A026767     else: return T(n-1,k-1) + T(n-1,k)
%o A026767 [sum(sum(T(j,k) for k in (0..n)) for j in (0..n)) for n in (0..30)] # _G. C. Greubel_, Oct 31 2019
%Y A026767 Cf. A026758, A026759, A026760, A026761, A026762, A026763, A026764, A026765, A026766, A026768.
%K A026767 nonn
%O A026767 0,2
%A A026767 _Clark Kimberling_