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A027049 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A027023.

%I A027049 #12 Nov 04 2019 19:39:15
%S A027049 16,120,952,7848,65580,550476,4631876,39047764,329784608,2790469092,
%T A027049 23656401612,200928615160,1709781846028,14575407966156,
%U A027049 124466311279620,1064636218853556,9120848372291680,78256468639080460,672393605270681188,5785139333187494936,49838058776228021388
%N A027049 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A027023.
%H A027049 G. C. Greubel, <a href="/A027049/b027049.txt">Table of n, a(n) for n = 3..1000</a>
%p A027049 T:= proc(n, k) option remember;
%p A027049       if k<3 or k=2*n then 1
%p A027049     else add(T(n-1, k-j), j=1..3)
%p A027049       fi
%p A027049     end:
%p A027049 seq(add(T(n,k)*T(n,k+3), k=0..2*n-3), n=3..30); # _G. C. Greubel_, Nov 04 2019
%t A027049 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+3], {k,0,2*n-3}], {n,3,30}] (* _G. C. Greubel_, Nov 04 2019 *)
%o A027049 (Sage)
%o A027049 @CachedFunction
%o A027049 def T(n, k):
%o A027049     if (k<3 or k==2*n): return 1
%o A027049     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027049 [sum(T(n,k)*T(n,k+3) for k in (0..2*n-3)) for n in (3..30)] # _G. C. Greubel_, Nov 04 2019
%K A027049 nonn
%O A027049 3,1
%A A027049 _Clark Kimberling_
%E A027049 More terms from _Sean A. Irvine_, Oct 22 2019