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A027274 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026552.

%I A027274 #14 Dec 19 2021 04:33:57
%S A027274 10,40,342,1279,11016,41462,359530,1365014,11899516,45501743,
%T A027274 398306769,1531614109,13450930624,51952990090,457449811458,
%U A027274 1773182087440,15646091896400,60825762159338,537651887201990,2095280066101886,18547910336883720,72432026278468535
%N A027274 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026552.
%H A027274 G. C. Greubel, <a href="/A027274/b027274.txt">Table of n, a(n) for n = 2..1000</a>
%F A027274 a(n) = Sum_{k=0..2*n-2} A026552(n,k) * A026552(n,k+2).
%t A027274 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 A027274 a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[n, k]*T[n, k+2], {k, 0, 2*n-2}]];
%t A027274 Table[a[n], {n,2,40}] (* _G. C. Greubel_, Dec 18 2021 *)
%o A027274 (Sage)
%o A027274 @CachedFunction
%o A027274 def T(n,k): # T = A026552
%o A027274     if (k==0 or k==2*n): return 1
%o A027274     elif (k==1 or k==2*n-1): return (n+2)//2
%o A027274     elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A027274     else: return T(n-1, k) + T(n-1, k-2)
%o A027274 @CachedFunction
%o A027274 def a(n): return sum( T(n,k)*T(n,k+2) for k in (0..2*n-2) )
%o A027274 [a(n) for n in (2..40)] # _G. C. Greubel_, Dec 18 2021
%Y A027274 Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027273, A027275, A027276.
%K A027274 nonn
%O A027274 2,1
%A A027274 _Clark Kimberling_
%E A027274 More terms from _Sean A. Irvine_, Oct 26 2019