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A027270 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026536.

%I A027270 #14 Apr 13 2022 01:15:52
%S A027270 2,10,104,420,3786,14826,131264,510576,4508580,17523506,154773696,
%T A027270 602175444,5323519838,20744201142,183586707648,716553432640,
%U A027270 6348284151024,24816637181076,220081449149440,861581808936200,7647723960962932,29978812970646870,266322435212031984
%N A027270 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026536.
%H A027270 G. C. Greubel, <a href="/A027270/b027270.txt">Table of n, a(n) for n = 2..500</a>
%F A027270 a(n) = Sum_{k=0..2n-3} A026536(n,k) * A026536(n,k+3).
%t A027270 T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n], T[n-1, k-2] +T[n-1, k-1] +T[n-1, k], T[n-1, k-2] +T[n-1, k]] ]];
%t A027270 Table[Sum[T[n,k]*T[n,k+3], {k,0,2*n-3}], {n,2,40}] (* _G. C. Greubel_, Apr 12 2022 *)
%o A027270 (SageMath)
%o A027270 @CachedFunction
%o A027270 def T(n, k): # A026536
%o A027270     if k == 0 or k == 2*n: return 1
%o A027270     elif k == 1 or k == 2*n-1: return n//2
%o A027270     elif n % 2 == 1: return T(n-1, k-2) + T(n-1, k)
%o A027270     return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
%o A027270 def A027270(n): return sum(T(n,k)*T(n,k+3) for k in (0..2*n-3))
%o A027270 [A027270(n) for n in (2..40)] # _G. C. Greubel_, Apr 12 2022
%Y A027270 Cf. A026536, A027267, A027268, A027269.
%K A027270 nonn
%O A027270 2,1
%A A027270 _Clark Kimberling_
%E A027270 More terms from _Sean A. Irvine_, Oct 26 2019
%E A027270 a(2) = 2 prepended by _G. C. Greubel_, Apr 12 2022