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A027079 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A027052.

%I A027079 #12 Nov 07 2019 21:11:26
%S A027079 0,4,24,160,1136,8420,64224,499984,3952928,31634724,255682432,
%T A027079 2083562368,17097573344,141143273396,1171240794072,9763809318912,
%U A027079 81724975129664,686539343850164
%N A027079 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A027052.
%H A027079 G. C. Greubel, <a href="/A027079/b027079.txt">Table of n, a(n) for n = 1..1000</a>
%p A027079 T:= proc(n, k) option remember;
%p A027079       if k<0 or k>2*n then 0
%p A027079     elif k=0 or k=2 or k=2*n then 1
%p A027079     elif k=1 then 0
%p A027079     else add(T(n-1, k-j), j=1..3)
%p A027079       fi
%p A027079     end:
%p A027079 seq( add(T(n,k)*T(n,k+1), k=0..2*n-1), n=1..30); # _G. C. Greubel_, Nov 07 2019
%t A027079 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]*T[n, k+1], {k, 0, 2*n-1}], {n,30}] (* _G. C. Greubel_, Nov 07 2019 *)
%o A027079 (Sage)
%o A027079 @CachedFunction
%o A027079 def T(n, k):
%o A027079     if (k<0 or k>2*n): return 0
%o A027079     elif (k==0 or k==2 or k==2*n): return 1
%o A027079     elif (k==1): return 0
%o A027079     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027079 [sum(T(n,k)*T(n,k+1) for k in (0..2*n-1)) for n in (1..30)] # _G. C. Greubel_, Nov 07 2019
%K A027079 nonn
%O A027079 1,2
%A A027079 _Clark Kimberling_