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A026756 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026747.

%I A026756 #10 Oct 29 2019 21:09:50
%S A026756 1,3,8,18,42,90,204,432,972,2052,4610,9726,21859,46125,103783,219099,
%T A026756 493699,1042899,2353716,4975350,11247138,23790714,53867343,114020601,
%U A026756 258571256,547672566,1243848888,2636201532,5995717860
%N A026756 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026747.
%H A026756 G. C. Greubel, <a href="/A026756/b026756.txt">Table of n, a(n) for n = 0..1000</a>
%p A026756 A026747 := proc(n,k) option remember;
%p A026756    if n<0 then 0;
%p A026756    elif k=0 or k = n then 1;
%p A026756    elif type(n,'even') and k <= n/2 then
%p A026756         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026756    else
%p A026756        procname(n-1,k-1)+procname(n-1,k) ;
%p A026756    end if ;
%p A026756 end proc:
%p A026756 seq(add(add(A026747(i,j), j=0..n), i=0..n), n=0..30); # _G. C. Greubel_, Oct 29 2019
%t A026756 T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[EvenQ[n] && 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] ]]]; Table[Sum[T[i, j], {i,0,n},{j,0,n}], {n,0,30}] (* _G. C. Greubel_, Oct 29 2019 *)
%o A026756 (Sage)
%o A026756 @CachedFunction
%o A026756 def T(n, k):
%o A026756     if (n<0): return 0
%o A026756     elif (k==0 or k==n): return 1
%o A026756     elif (mod(n,2)==0 and k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
%o A026756     else: return T(n-1,k-1) + T(n-1,k)
%o A026756 [sum(sum(T(i,j) for j in (0..n)) for i in (0..n)) for n in (0..30)] # _G. C. Greubel_, Oct 29 2019
%Y A026756 Cf. A026747, A026748, A026749, A026750, A026751, A026752, A026753, A026754, A026755, A026757.
%K A026756 nonn
%O A026756 0,2
%A A026756 _Clark Kimberling_