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A027275 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026552.

%I A027275 #14 Dec 19 2021 04:33:48
%S A027275 24,232,954,8560,33648,297940,1159844,10242416,39809076,351561242,
%T A027275 1367463642,12086555584,47082494816,416589513644,1625447736120,
%U A027275 14397549291280,56265306436584,498879779964188,1952476424575980,17327820010494464,67907006619888744
%N A027275 a(n) = Sum_{k=0..2n-3} T(n,k) * T(n,k+3), with T given by A026552.
%H A027275 G. C. Greubel, <a href="/A027275/b027275.txt">Table of n, a(n) for n = 3..1000</a>
%F A027275 a(n) = Sum_{k=0..2*n-3} A026552(n, k) * A026552(n, k+3).
%t A027275 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 A027275 a[n_]:= a[n]= Block[{$RecursionLimit = Infinity}, Sum[T[n, k]*T[n, k+3], {k, 0, 2*n-3}]];
%t A027275 Table[a[n], {n,3,40}] (* _G. C. Greubel_, Dec 18 2021 *)
%o A027275 (Sage)
%o A027275 @CachedFunction
%o A027275 def T(n,k): # T = A026552
%o A027275     if (k==0 or k==2*n): return 1
%o A027275     elif (k==1 or k==2*n-1): return (n+2)//2
%o A027275     elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A027275     else: return T(n-1, k) + T(n-1, k-2)
%o A027275 @CachedFunction
%o A027275 def a(n): return sum( T(n,k)*T(n,k+3) for k in (0..2*n-3) )
%o A027275 [a(n) for n in (3..40)] # _G. C. Greubel_, Dec 18 2021
%Y A027275 Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027273, A027274, A027276.
%K A027275 nonn
%O A027275 3,1
%A A027275 _Clark Kimberling_
%E A027275 More terms from _Sean A. Irvine_, Oct 26 2019