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A027035 a(n) = Sum_{k=0..n} T(n,n+k), T given by A027023.

%I A027035 #11 Nov 04 2019 19:38:45
%S A027035 1,2,5,14,39,112,323,932,2693,7790,22565,65466,190243,553748,1614363,
%T A027035 4713432,13780841,40343210,118243273,346937614,1018958151,2995407840,
%U A027035 8812890391,25948662684,76457517949,225429675606,665069604713
%N A027035 a(n) = Sum_{k=0..n} T(n,n+k), T given by A027023.
%H A027035 G. C. Greubel, <a href="/A027035/b027035.txt">Table of n, a(n) for n = 0..1000</a>
%p A027035 T:= proc(n, k) option remember;
%p A027035       if k<3 or k=2*n then 1
%p A027035     else add(T(n-1, k-j), j=1..3)
%p A027035       fi
%p A027035     end:
%p A027035 seq(add(T(n, k), k=n..2*n), n=0..30); # _G. C. Greubel_, Nov 04 2019
%t A027035 T[n_, k_]:= T[n, k]= If[k<3 || k==2*n, 1, Sum[T[n-1, k-j], {j,3}]]; Table[Sum[T[n,k], {k,n,2*n}], {n,0,30}] (* _G. C. Greubel_, Nov 04 2019 *)
%o A027035 (Sage)
%o A027035 @CachedFunction
%o A027035 def T(n, k):
%o A027035     if (k<3 or k==2*n): return 1
%o A027035     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027035 [sum(T(n, k) for k in (n..2*n)) for n in (0..30)] # _G. C. Greubel_, Nov 04 2019
%K A027035 nonn
%O A027035 0,2
%A A027035 _Clark Kimberling_