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A027077 a(n) = Sum_{k=n+1..2*n} T(n,k), T given by A027052.

%I A027077 #8 Nov 07 2019 08:28:28
%S A027077 1,3,8,24,71,209,612,1784,5189,15081,43838,127528,371395,1082951,
%T A027077 3161866,9243400,27055153,79280601,232567194,682905120,2007104343,
%U A027077 5904004451,17380510458,51202600920,150942696637,445247984543
%N A027077 a(n) = Sum_{k=n+1..2*n} T(n,k), T given by A027052.
%H A027077 G. C. Greubel, <a href="/A027077/b027077.txt">Table of n, a(n) for n = 1..1000</a>
%p A027077 T:= proc(n, k) option remember;
%p A027077       if k<0 or k>2*n then 0
%p A027077     elif k=0 or k=2 or k=2*n then 1
%p A027077     elif k=1 then 0
%p A027077     else add(T(n-1, k-j), j=1..3)
%p A027077       fi
%p A027077     end:
%p A027077 seq( add(T(n,k), k=n+1..2*n), n=1..30); # _G. C. Greubel_, Nov 06 2019
%t A027077 T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[T[n, k], {k, n+1, 2*n}], {n, 1, 30}] (* _G. C. Greubel_, Nov 06 2019 *)
%o A027077 (Sage)
%o A027077 @CachedFunction
%o A027077 def T(n, k):
%o A027077     if (k<0 or k>2*n): return 0
%o A027077     elif (k==0 or k==2 or k==2*n): return 1
%o A027077     elif (k==1): return 0
%o A027077     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027077 [sum(T(n,k) for k in (n+1..2*n)) for n in (1..30)] # _G. C. Greubel_, Nov 06 2019
%K A027077 nonn
%O A027077 1,2
%A A027077 _Clark Kimberling_
%E A027077 Offset changed by _G. C. Greubel_, Nov 06 2019