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A027218 a(n) = Sum_{k=0..n-3} T(n,k)*T(n,k+3), T given by A026736.

%I A027218 #16 Sep 29 2024 17:00:47
%S A027218 1,9,51,279,1277,6235,26789,125370,525082,2409886,9969722,45289767,
%T A027218 186105280,840402559,3439358196,15472942142,63155131233,283400162019
%N A027218 a(n) = Sum_{k=0..n-3} T(n,k)*T(n,k+3), T given by A026736.
%H A027218 G. C. Greubel, <a href="/A027218/b027218.txt">Table of n, a(n) for n = 3..1000</a>
%t A027218 T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[EvenQ[n] && k==(n-2)/2, T[n-1,k-1] + T[n-2,k-1] + T[n-1,k], T[n-1,k-1] + T[n-1,k]]]; Table[Sum[T[n,k]*T[n,k+3], {k, 0, n-3}], {n, 3, 30}] (* _G. C. Greubel_, Jul 19 2019 *)
%o A027218 (PARI) T(n, k) = if(k==n || k==0, 1, k==n-1, n, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
%o A027218 for(n=3,20, print1(sum(k=0, n-3, T(n, k)*T(n,k+3)), ", ")) \\ _G. C. Greubel_, Jul 19 2019
%o A027218 (Sage)
%o A027218 @CachedFunction
%o A027218 def T(n, k):
%o A027218     if (k==0 or k==n): return 1
%o A027218     elif (mod(n, 2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
%o A027218     else: return T(n-1, k-1) + T(n-1, k)
%o A027218 [sum(T(n,k)*T(n,k+3) for k in (0..n-3)) for n in (3..30)] # _G. C. Greubel_, Jul 19 2019
%o A027218 (GAP)
%o A027218 T:= function(n, k)
%o A027218     if k=0 or k=n then return 1;
%o A027218     elif k=n-1 then return n;
%o A027218     elif (n mod 2)=0 and k=Int((n-2)/2) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
%o A027218     else return T(n-1, k-1) + T(n-1, k);
%o A027218     fi;
%o A027218   end;
%o A027218 List([3..20], n-> Sum([0..n-3], k-> T(n, k)*T(n,k+3) )); # _G. C. Greubel_, Jul 19 2019
%Y A027218 Cf. A026736.
%K A027218 nonn
%O A027218 3,2
%A A027218 _Clark Kimberling_