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A027985 a(n) = Sum_{k=0..2*n-1} T(n, k)*T(n, k+1), T given by A027960.

%I A027985 #11 Jun 13 2025 07:44:21
%S A027985 6,35,144,564,2186,8468,32856,127729,497454,1940525,7580656,29651385,
%T A027985 116111194,455138499,1785707924,7011933544,27554583254,108355491404,
%U A027985 426368213364,1678704356644,6613026412314,26064305550054,102777232982624
%N A027985 a(n) = Sum_{k=0..2*n-1} T(n, k)*T(n, k+1), T given by A027960.
%H A027985 G. C. Greubel, <a href="/A027985/b027985.txt">Table of n, a(n) for n = 1..1000</a>
%F A027985 a(n) = A360278(n) - 1 + Sum_{k=n..2*n-1} A027960(n,k)*A027960(n,k+1), for n >= 1. - _G. C. Greubel_, Jun 13 2025
%t A027985 f[n_, k_]:= f[n,k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
%t A027985 A027960[n_, k_]:= LucasL[k+1] - f[n,k]*Boole[k>n];
%t A027985 A027985[n_]:= A027985[n]= Sum[A027960[n,k]*A027960[n,k+1], {k,0,2*n-1}];
%t A027985 Table[A027985[n], {n,40}] (* _G. C. Greubel_, Jun 13 2025 *)
%o A027985 (Magma)
%o A027985 f:= func< n,k | (&+[Binomial(2*n-k+j,j)*Lucas(2*(k-n-j)): j in [0..k-n-1]]) >;
%o A027985 A027960:= func< n,k | k le n select Lucas(k+1) else Lucas(k+1) - f(n,k) >;
%o A027985 A027985:= func< n | (&+[A027960(n,k)*A027960(n,k+1): k in [0..2*n-1]]) >;
%o A027985 [A027985(n): n in [1..40]]; // _G. C. Greubel_, Jun 13 2025
%o A027985 (SageMath)
%o A027985 def L(n): return lucas_number2(n,1,-1)
%o A027985 def f(n,k): return sum(binomial(2*n-k+j,j)*L(2*(k-n-j)) for j in range(k-n))
%o A027985 def A027960(n,k): return L(k+1) - f(n,k)*int(k>n)
%o A027985 def A027985(n): return sum(A027960(n,k)*A027960(n,k+1) for k in range(2*n))
%o A027985 print([A027985(n) for n in range(1,41)]) # _G. C. Greubel_, Jun 13 2025
%Y A027985 Cf. A027960, A360278.
%K A027985 nonn
%O A027985 1,1
%A A027985 _Clark Kimberling_