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A026789 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026780.

%I A026789 #11 Feb 10 2022 09:18:27
%S A026789 1,3,8,19,45,103,239,545,1262,2887,6700,15397,35848,82757,193320,
%T A026789 448175,1050217,2443963,5743267,13410053,31593029,73984575,174689181,
%U A026789 410141597,970289011,2283205051,5410611863,12756825609,30274963923
%N A026789 a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026780.
%H A026789 G. C. Greubel, <a href="/A026789/b026789.txt">Table of n, a(n) for n = 0..1000</a>
%p A026789 T:= proc(n,k) option remember;
%p A026789     if n<0 then 0;
%p A026789     elif k=0 or k =n then 1;
%p A026789     elif k <= n/2 then
%p A026789         procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026789     else
%p A026789         procname(n-1,k-1)+procname(n-1,k) ;
%p A026789     fi ;
%p A026789 end proc:
%p A026789 seq( add(add(T(j,k), k=0..n), j=0..n), n=0..30); # _G. C. Greubel_, Nov 02 2019
%t A026789 T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[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] ]]];
%t A026789 Table[Sum[T[j, k], {k, 0, n}, {j, 0, n}], {n,0,30}] (* _G. C. Greubel_, Nov 02 2019 *)
%o A026789 (Sage)
%o A026789 @CachedFunction
%o A026789 def T(n, k):
%o A026789     if (n<0): return 0
%o A026789     elif (k==0 or k==n): return 1
%o A026789     elif (k<=n/2): return T(n-1,k-1) + T(n-2,k-1) + T(n-1,k)
%o A026789     else: return T(n-1,k-1) + T(n-1,k)
%o A026789 [sum( sum( T(j,k) for k in (0..n)) for j in (0..n)) for n in (0..30)] # _G. C. Greubel_, Nov 02 2019
%Y A026789 Partial sums of A026787.
%Y A026789 Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026788, A026790.
%K A026789 nonn
%O A026789 0,2
%A A026789 _Clark Kimberling_