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A027041 a(n) = Sum_{k=0..n} T(n,k) * T(n,2n-k), with T given by A027023.

%I A027041 #16 Jun 27 2022 21:18:55
%S A027041 1,2,5,20,77,306,1209,4656,17713,66618,248025,916020,3359789,12250026,
%T A027041 44435997,160466304,577185745,2068826290,7392167585,26338879556,
%U A027041 93609302941,331924381218,1174482354493,4147807582672,14622567051025,51466158436298,180869949252245,634753692067716
%N A027041 a(n) = Sum_{k=0..n} T(n,k) * T(n,2n-k), with T given by A027023.
%H A027041 G. C. Greubel, <a href="/A027041/b027041.txt">Table of n, a(n) for n = 0..1000</a>
%p A027041 T:= proc(n, k) option remember;
%p A027041       if k<3 or k=2*n then 1
%p A027041     else add(T(n-1, k-j), j=1..3)
%p A027041       fi
%p A027041     end:
%p A027041 seq(add(T(n, k)*T(n,2*n-k), k=0..n), n=0..30); # _G. C. Greubel_, Nov 04 2019
%t A027041 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,2*n-k], {k,0,n}], {n,0,30}] (* _G. C. Greubel_, Nov 04 2019 *)
%o A027041 (Sage)
%o A027041 @CachedFunction
%o A027041 def T(n, k):
%o A027041     if (k<3 or k==2*n): return 1
%o A027041     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027041 [sum(T(n, k)*T(n,2*n-k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Nov 04 2019
%K A027041 nonn
%O A027041 0,2
%A A027041 _Clark Kimberling_
%E A027041 More terms from _Sean A. Irvine_, Oct 22 2019