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A027047 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A027023.

%I A027047 #13 Nov 05 2019 01:01:28
%S A027047 2,8,50,336,2418,18088,138850,1086016,8617122,69159896,560290322,
%T A027047 4574820624,37603654098,310873702392,2582964183874,21556333188288,
%U A027047 180609299685954,1518572497996568,12808849866774002,108351496132761104,918964407713589618,7812768025080427672
%N A027047 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A027023.
%H A027047 G. C. Greubel, <a href="/A027047/b027047.txt">Table of n, a(n) for n = 1..1000</a>
%p A027047 T:= proc(n, k) option remember;
%p A027047       if k<3 or k=2*n then 1
%p A027047     else add(T(n-1, k-j), j=1..3)
%p A027047       fi
%p A027047     end:
%p A027047 seq(add(T(n,k)*T(n,k+1), k=0..2*n-1), n=1..30); # _G. C. Greubel_, Nov 04 2019
%t A027047 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]*T[n,k+1], {k,0,2*n-1}], {n,1,30}] (* _G. C. Greubel_, Nov 04 2019 *)
%o A027047 (Sage)
%o A027047 @CachedFunction
%o A027047 def T(n, k):
%o A027047     if (k<3 or k==2*n): return 1
%o A027047     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027047 [sum(T(n,k)*T(n,k+1) for k in (0..2*n-1)) for n in (1..30)] # _G. C. Greubel_, Nov 04 2019
%K A027047 nonn
%O A027047 1,1
%A A027047 _Clark Kimberling_
%E A027047 More terms from _Sean A. Irvine_, Oct 22 2019