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A027273 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026552.

%I A027273 #14 Dec 19 2021 04:30:22
%S A027273 2,16,52,428,1516,12792,46936,402164,1504432,13015480,49288856,
%T A027273 429204354,1639174304,14340670000,55108565584,483825847108,
%U A027273 1868067054968,16445659005424,63734526307552,562323306397388,2185849699156352,19320211642880176,75288454939134992
%N A027273 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026552.
%H A027273 G. C. Greubel, <a href="/A027273/b027273.txt">Table of n, a(n) for n = 1..1000</a>
%F A027273 a(n) = Sum_{k=0..2*n-1} A026552(n, k)*A026552(n, k+1).
%t A027273 T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+2)/2], If[EvenQ[n], T[n-1, k-2] + T[n-1, k] + T[n-1, k-1], T[n-1, k-2] + T[n-1, k]]]]; (* T=A026552 *)
%t A027273 a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[n, k]*T[n, k+1], {k, 0, 2*n-1}]];
%t A027273 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Dec 18 2021 *)
%o A027273 (Sage)
%o A027273 @CachedFunction
%o A027273 def T(n,k): # T = A026552
%o A027273     if (k==0 or k==2*n): return 1
%o A027273     elif (k==1 or k==2*n-1): return (n+2)//2
%o A027273     elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A027273     else: return T(n-1, k) + T(n-1, k-2)
%o A027273 @CachedFunction
%o A027273 def a(n): return sum( T(n,k)*T(n,k+1) for k in (0..2*n-1) )
%o A027273 [a(n) for n in (1..40)] # _G. C. Greubel_, Dec 18 2021
%Y A027273 Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027274, A027275, A027276.
%K A027273 nonn
%O A027273 1,1
%A A027273 _Clark Kimberling_
%E A027273 More terms from _Sean A. Irvine_, Oct 26 2019