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A027080 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A027052.

%I A027080 #12 Nov 07 2019 21:11:32
%S A027080 2,15,100,757,5902,46907,377520,3065809,25078650,206416795,1708129244,
%T A027080 14202265321,118585167502,993915161547,8358970631568,70518298143329,
%U A027080 596590060985546,5060232622624651,43022268222676124,366575545244139845,3129747701356459022,26771150349554898415
%N A027080 a(n) = Sum_{k=0..2n-2} T(n,k) * T(n,k+2), with T given by A027052.
%H A027080 G. C. Greubel, <a href="/A027080/b027080.txt">Table of n, a(n) for n = 2..1000</a>
%p A027080 T:= proc(n, k) option remember;
%p A027080       if k<0 or k>2*n then 0
%p A027080     elif k=0 or k=2 or k=2*n then 1
%p A027080     elif k=1 then 0
%p A027080     else add(T(n-1, k-j), j=1..3)
%p A027080       fi
%p A027080     end:
%p A027080 seq( add(T(n,k)*T(n,k+2), k=0..2*n-2), n=2..30); # _G. C. Greubel_, Nov 07 2019
%t A027080 T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[Sum[T[n, k]*T[n, k+2], {k, 0, 2*n-2}], {n,2,30}] (* _G. C. Greubel_, Nov 07 2019 *)
%o A027080 (Sage)
%o A027080 @CachedFunction
%o A027080 def T(n, k):
%o A027080     if (k<0 or k>2*n): return 0
%o A027080     elif (k==0 or k==2 or k==2*n): return 1
%o A027080     elif (k==1): return 0
%o A027080     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027080 [sum(T(n,k)*T(n,k+2) for k in (0..2*n-2)) for n in (2..30)] # _G. C. Greubel_, Nov 07 2019
%K A027080 nonn
%O A027080 2,1
%A A027080 _Clark Kimberling_
%E A027080 More terms from _Sean A. Irvine_, Oct 22 2019