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A027217 a(n) = Sum_{k=0..n-2} T(n,k)*T(n,k+2), T given by A026736.

%I A027217 #12 Sep 29 2024 17:00:35
%S A027217 1,6,32,136,640,2593,11860,47532,215531,861334,3893621,15549166,
%T A027217 70199065,280316029,1264697307,5050617474,22776900816,90972831448,
%U A027217 410117333080
%N A027217 a(n) = Sum_{k=0..n-2} T(n,k)*T(n,k+2), T given by A026736.
%H A027217 G. C. Greubel, <a href="/A027217/b027217.txt">Table of n, a(n) for n = 2..1000</a>
%t A027217 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+2], {k, 0, n-2}], {n, 2, 30}] (* _G. C. Greubel_, Jul 19 2019 *)
%o A027217 (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 A027217 vector(20, n, n++; sum(k=0, n-2, T(n, k)*T(n,k+2)) ) \\ _G. C. Greubel_, Jul 19 2019
%o A027217 (Sage)
%o A027217 @CachedFunction
%o A027217 def T(n, k):
%o A027217     if (k==0 or k==n): return 1
%o A027217     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 A027217     else: return T(n-1, k-1) + T(n-1, k)
%o A027217 [sum(T(n,k)*T(n,k+2) for k in (0..n-2)) for n in (2..30)] # _G. C. Greubel_, Jul 19 2019
%o A027217 (GAP)
%o A027217 T:= function(n, k)
%o A027217     if k=0 or k=n then return 1;
%o A027217     elif k=n-1 then return n;
%o A027217     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 A027217     else return T(n-1, k-1) + T(n-1, k);
%o A027217     fi;
%o A027217   end;
%o A027217 List([2..20], n-> Sum([0..n-2], k-> T(n, k)*T(n,k+2) )); # _G. C. Greubel_, Jul 19 2019
%Y A027217 Cf. A026736.
%K A027217 nonn
%O A027217 2,2
%A A027217 _Clark Kimberling_