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A027263 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026519.

%I A027263 #16 Dec 22 2021 07:03:45
%S A027263 2,6,52,180,1516,5502,46936,174456,1504432,5673140,49288856,187675644,
%T A027263 1639174304,6284986554,55108565584,212408191568,1868067054968,
%U A027263 7229648901024,63734526307552,247468885359240,2185849699156352,8510025522045036,75288454939134992,293772371437293720
%N A027263 a(n) = Sum_{k=0..2n-1} T(n,k) * T(n,k+1), with T given by A026519.
%H A027263 G. C. Greubel, <a href="/A027263/b027263.txt">Table of n, a(n) for n = 1..1000</a>
%F A027263 a(n) = Sum_{k=0..2n-1} A026519(n,k) * A026519(n,k+1).
%t A027263 T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[(n+1)/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 = A026519 *)
%t A027263 a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[n, k]*T[n, k+1], {k, 0, 2*n-1}] ];
%t A027263 Table[a[n], {n, 40}] (* _G. C. Greubel_, Dec 21 2021 *)
%o A027263 (Sage)
%o A027263 @CachedFunction
%o A027263 def T(n,k): # T = A026519
%o A027263     if (k<0 or k>2*n): return 0
%o A027263     elif (k==0 or k==2*n): return 1
%o A027263     elif (k==1 or k==2*n-1): return (n+1)//2
%o A027263     elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
%o A027263     else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A027263 @CachedFunction
%o A027263 def a(n): return sum( T(n,k)*T(n,k+1) for k in (0..2*n-1) )
%o A027263 [a(n) for n in (1..40)] # _G. C. Greubel_, Dec 21 2021
%Y A027263 Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027262, A027264, A027265, A027266.
%K A027263 nonn
%O A027263 1,1
%A A027263 _Clark Kimberling_
%E A027263 More terms from _Sean A. Irvine_, Oct 26 2019