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A027074 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,2n-k), with T given by A027052.

%I A027074 #13 Nov 07 2019 08:28:43
%S A027074 1,1,4,22,93,389,1570,6144,23629,89551,335430,1244762,4583293,
%T A027074 16765087,60980096,220724896,795540601,2856541663,10222762962,
%U A027074 36475315442,129796579409,460757642587,1632012075912,5768986242408
%N A027074 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,2n-k), with T given by A027052.
%H A027074 G. C. Greubel, <a href="/A027074/b027074.txt">Table of n, a(n) for n = 1..1000</a>
%p A027074 T:= proc(n, k) option remember;
%p A027074       if k<0 or k>2*n then 0
%p A027074     elif k=0 or k=2 or k=2*n then 1
%p A027074     elif k=1 then 0
%p A027074     else add(T(n-1, k-j), j=1..3)
%p A027074       fi
%p A027074     end:
%p A027074 seq( add(T(n,k)*T(n,2*n-k), k=0..n-1), n=1..30); # _G. C. Greubel_, Nov 06 2019
%t A027074 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,2*n-k], {k,0,n-1}], {n, 30}] (* _G. C. Greubel_, Nov 06 2019 *)
%o A027074 (Sage)
%o A027074 @CachedFunction
%o A027074 def T(n, k):
%o A027074     if (k<0 or k>2*n): return 0
%o A027074     elif (k==0 or k==2 or k==2*n): return 1
%o A027074     elif (k==1): return 0
%o A027074     else: return sum(T(n-1, k-j) for j in (1..3))
%o A027074 [sum(T(n,k)*T(n,2*n-k) for k in (0..n-1)) for n in (1..30)] # _G. C. Greubel_, Nov 06 2019
%K A027074 nonn
%O A027074 1,3
%A A027074 _Clark Kimberling_