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A026778 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026769.

%I A026778 #12 Mar 19 2020 14:09:47
%S A026778 1,3,7,16,35,78,171,383,849,1919,4301,9807,22193,50993,116349,269094,
%T A026778 618277,1437916,3323277,7764996,18035275,42304904,98668223,232198092,
%U A026778 543453693,1282401208,3010275001,7119589730,16753996391
%N A026778 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026769.
%H A026778 G. C. Greubel, <a href="/A026778/b026778.txt">Table of n, a(n) for n = 0..1000</a>
%p A026778 T:= proc(n,k) option remember;
%p A026778    if n<0 then 0;
%p A026778    elif k=0 or k=n then 1;
%p A026778    elif n=2 and k=1 then 2;
%p A026778    elif k <= (n-1)/2 then
%p A026778         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026778    else
%p A026778        procname(n-1,k-1)+procname(n-1,k) ;
%p A026778    end if ;
%p A026778 end proc;
%p A026778 seq(add(add(T(j,k), k=0..n), j=0..n), n=0..30); # _G. C. Greubel_, Nov 01 2019
%t A026778 T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[n==2 && k==1, 2, If[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_, Nov 01 2019 *)
%o A026778 (Sage)
%o A026778 @CachedFunction
%o A026778 def T(n, k):
%o A026778     if n < 0:
%o A026778         return 0
%o A026778     elif (k==0 or k==n): return 1
%o A026778     elif (n==2 and k==1): return 2
%o A026778     elif (k<=(n-1)/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
%o A026778     else: return T(n-1,k-1) + T(n-1,k)
%o A026778 [sum(sum(T(j,k) for k in (0..n)) for j in (0..n)) for n in (0..30)] # _G. C. Greubel_, Nov 01 2019
%Y A026778 Cf. A026769, A026770, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026779.
%K A026778 nonn
%O A026778 0,2
%A A026778 _Clark Kimberling_