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A027986 a(n) = Sum_{k=0..2*n-2} T(n, k)*T(n, k+2), T given by A027960.

%I A027986 #10 Jun 14 2025 04:20:57
%S A027986 20,100,420,1694,6746,26735,105722,417613,1648692,6507197,25681362,
%T A027986 101359219,400094756,1579568360,6237401648,24635828774,97327071806,
%U A027986 384596629610,1520138671688,6009885957464,23765835521966,94003008751940
%N A027986 a(n) = Sum_{k=0..2*n-2} T(n, k)*T(n, k+2), T given by A027960.
%H A027986 G. C. Greubel, <a href="/A027986/b027986.txt">Table of n, a(n) for n = 2..1000</a>
%F A027986 a(n) = A000032(2*n+1) - 7 + 3*(n mod 2) + Sum_{k=n-1..2*n-2} A027960(n, k)*A027960(n, k+2). - _G. C. Greubel_, Jun 14 2025
%t A027986 f[n_, k_]:= f[n,k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
%t A027986 A027960[n_, k_]:= LucasL[k+1] -f[n,k]*Boole[k>n];
%t A027986 A027986[n_]:= A027986[n]= Sum[A027960[n,k]*A027960[n,k+2], {k,0,2*n-2}];
%t A027986 Table[A027986[n], {n,2,40}] (* _G. C. Greubel_, Jun 14 2025 *)
%o A027986 (Magma)
%o A027986 f:= func< n,k | (&+[Binomial(2*n-k+j,j)*Lucas(2*(k-n-j)): j in [0..k-n-1]]) >;
%o A027986 A027960:= func< n,k | k le n select Lucas(k+1) else Lucas(k+1) - f(n,k) >;
%o A027986 A027986:= func< n | (&+[A027960(n,k)*A027960(n,k+2): k in [0..2*n-2]]) >;
%o A027986 [A027986(n): n in [2..40]]; // _G. C. Greubel_, Jun 14 2025
%o A027986 (SageMath)
%o A027986 @CachedFunction
%o A027986 def T(n, k): # T = A027960
%o A027986     if (k>2*n): return 0
%o A027986     elif (k<n+1): return lucas_number2(k+1,1,-1)
%o A027986     else: return T(n-1, k-2) + T(n-1, k-1)
%o A027986 def A027986(n): return sum(T(n,k)*T(n,k+2) for k in range(2*n-1))
%o A027986 print([A027986(n) for n in range(2,41)]) # _G. C. Greubel_, Jun 14 2025
%Y A027986 Cf. A027960.
%K A027986 nonn
%O A027986 2,1
%A A027986 _Clark Kimberling_