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A027268 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026536.

%I A027268 #12 Apr 12 2022 04:23:11
%S A027268 0,6,20,180,644,5502,20292,174456,654632,5673140,21528000,187675644,
%T A027268 717800628,6284986554,24178479500,212408191568,820811282352,
%U A027268 7229648901024,28037230854096,247468885359240,962488105227160,8510025522045036,33177800527098040,293772371437293720
%N A027268 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026536.
%H A027268 G. C. Greubel, <a href="/A027268/b027268.txt">Table of n, a(n) for n = 1..500</a>
%F A027268 a(n) = Sum_{k=0..2n-1} A026536(n,k) * A026536(n,k+1)
%t A027268 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 A027268 Table[Sum[T[n, k]*T[n,k+1], {k,0,2*n-1}], {n,40}] (* _G. C. Greubel_, Apr 12 2022 *)
%o A027268 (SageMath)
%o A027268 @CachedFunction
%o A027268 def T(n, k): # A026536
%o A027268     if k < 0 or n < 0: return 0
%o A027268     elif k == 0 or k == 2*n: return 1
%o A027268     elif k == 1 or k == 2*n-1: return n//2
%o A027268     elif n % 2 == 1: return T(n-1, k-2) + T(n-1, k)
%o A027268     return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
%o A027268 def A027268(n): return sum(T(n,k)*T(n,k+1) for k in (0..2*n-1))
%o A027268 [A027268(n) for n in (1..40)] # _G. C. Greubel_, Apr 12 2022
%Y A027268 Cf. A026536, A027267, A027269.
%K A027268 nonn
%O A027268 1,2
%A A027268 _Clark Kimberling_
%E A027268 More terms from _Sean A. Irvine_, Oct 26 2019