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A027984 a(n) = Sum_{k=0..n} T(n, k)*T(n, n+k), T given by A027960.

%I A027984 #15 Jun 10 2025 08:58:42
%S A027984 1,6,20,58,161,436,1165,3088,8146,21426,56255,147538,386681,1013026,
%T A027984 2653240,6948058,18193141,47634936,124717445,326526748,854877926,
%U A027984 2238131506,5859556195,15340601158,40162350961,105146619486,275277778940
%N A027984 a(n) = Sum_{k=0..n} T(n, k)*T(n, n+k), T given by A027960.
%H A027984 G. C. Greubel, <a href="/A027984/b027984.txt">Table of n, a(n) for n = 0..1000</a>
%H A027984 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-2,1).
%F A027984 Conjectures from _Colin Barker_, Feb 25 2015: (Start)
%F A027984 a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4).
%F A027984 G.f.: (1-x)*(1+x)*(1+2*x) / ((1-3*x+x^2)*(1-x-x^2)). (End)
%F A027984 a(n) = (1/2)*(5*A000045(2*n+2) + A000032(2*n+1) - A000032(n+3)). - _G. C. Greubel_, Jun 10 2025
%t A027984 LinearRecurrence[{4,-3,-2,1}, {1,6,20,58}, 41] (* _G. C. Greubel_, Jun 10 2025 *)
%o A027984 (PARI) T027960(r,n) = if(r<0||n>2*r, return(0)); if(n==0||n==2*r, return(1)); if(n==1, 3, T027960(r-1, n-1)+T027960(r-1, n-2));
%o A027984 a(n) = sum(k=0, n, T027960(n, k)*T027960(n, n+k)); \\ _Michel Marcus_, Feb 25 2015
%o A027984 (Magma)
%o A027984 A027984:= func< n | (5*Fibonacci(2*n+2) +Lucas(2*n+1) -Lucas(n+3))/2 >;
%o A027984 [A027984(n): n in [0..40]]; // _G. C. Greubel_, Jun 10 2025
%o A027984 (SageMath)
%o A027984 L = lucas_number2
%o A027984 def A027984(n): return (5*fibonacci(2*n+2) +L(2*n+1,1,-1) -L(n+3,1,-1))//2
%o A027984 print([A027984(n) for n in range(41)]) # _G. C. Greubel_, Jun 10 2025
%Y A027984 Cf. A000032, A000045, A027960.
%K A027984 nonn
%O A027984 0,2
%A A027984 _Clark Kimberling_