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A027264 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026519.

%I A027264 #14 Dec 22 2021 07:04:02
%S A027264 5,40,150,1279,4797,41462,156900,1365014,5205950,45501743,174609162,
%T A027264 1531614109,5906040623,51952990090,201114700568,1773182087440,
%U A027264 6885880226784,60825762159338,236826459554380,2095280066101886,8175978023317170,72432026278468535,283166067626865540
%N A027264 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A026519.
%H A027264 G. C. Greubel, <a href="/A027264/b027264.txt">Table of n, a(n) for n = 2..1000</a>
%F A027264 a(n) = Sum_{k=0..2n-2} A026519(n,k) * A026519(n,k+2).
%t A027264 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 A027264 a[n_] := a[n] = Block[{$RecursionLimit = Infinity}, Sum[T[n, k]*T[n, k+2], {k, 0, 2*n-2}] ];
%t A027264 Table[a[n], {n, 2, 40}] (* _G. C. Greubel_, Dec 21 2021 *)
%o A027264 (Sage)
%o A027264 @CachedFunction
%o A027264 def T(n,k): # T = A026519
%o A027264     if (k<0 or k>2*n): return 0
%o A027264     elif (k==0 or k==2*n): return 1
%o A027264     elif (k==1 or k==2*n-1): return (n+1)//2
%o A027264     elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
%o A027264     else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o A027264 @CachedFunction
%o A027264 def a(n): return sum( T(n,k)*T(n,k+2) for k in (0..2*n-2) )
%o A027264 [a(n) for n in (2..40)] # _G. C. Greubel_, Dec 21 2021
%Y A027264 Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027262, A027263, A027265, A027266.
%K A027264 nonn
%O A027264 2,1
%A A027264 _Clark Kimberling_
%E A027264 More terms from _Sean A. Irvine_, Oct 26 2019