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A027269 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026536.

%I A027269 #14 Apr 12 2022 04:19:56
%S A027269 1,5,19,150,561,4797,18089,156900,596674,5205950,19932353,174609162,
%T A027269 672106267,5906040623,22829936683,201114700568,780077588440,
%U A027269 6885880226784,26784015828458,236826459554380,923352937530146,8175978023317170,31940549289135429,283166067626865540
%N A027269 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026536.
%H A027269 G. C. Greubel, <a href="/A027269/b027269.txt">Table of n, a(n) for n = 1..500</a>
%F A027269 a(n) = Sum_{k=0..2n-2} A026536(n,k) * A026536(n,k+2).
%t A027269 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 A027269 Table[Sum[T[n,k]*T[n,k+2], {k,0,2*n-2}], {n,40}] (* _G. C. Greubel_, Apr 12 2022 *)
%o A027269 (SageMath)
%o A027269 @CachedFunction
%o A027269 def T(n, k): # A026536
%o A027269     if k < 0 or n < 0: return 0
%o A027269     elif k == 0 or k == 2*n: return 1
%o A027269     elif k == 1 or k == 2*n-1: return n//2
%o A027269     elif n % 2 == 1: return T(n-1, k-2) + T(n-1, k)
%o A027269     return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
%o A027269 def A027269(n): return sum(T(n,k)*T(n,k+2) for k in (0..2*n-2))
%o A027269 [A027269(n) for n in (1..40)] # _G. C. Greubel_, Apr 12 2022
%Y A027269 Cf. A026536, A027267, A027268, A027270.
%K A027269 nonn
%O A027269 1,2
%A A027269 _Clark Kimberling_
%E A027269 More terms from _Sean A. Irvine_, Oct 26 2019
%E A027269 a(1) = 1 prepended by _G. C. Greubel_, Apr 12 2022