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%I #18 Oct 21 2022 22:11:20
%S 1,6,27,72,270,648,2268,5184,17496,38880,128304,279936,909792,1959552,
%T 6298560,13436928,42830208,90699264,287214336,604661760,1904684544,
%U 3990767616,12516498432,26121388032,81629337600,169789022208
%N a(n) = Sum_{k=0..2n} (k+1) * A026552(n, k).
%H G. C. Greubel, <a href="/A027276/b027276.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,12,0,-36).
%F a(n) = Sum_{k=0..2n} (k+1) * A026552(n, k).
%F G.f.: (1 +6*x +15*x^2 -18*x^3)/(1-6*x^2)^2.
%F a(n) = -(1/2)*[n=0] + (1/4)*6^(n/2)*(n + 1)*(3*(1 + (-1)^n) + sqrt(6)*(1 - (-1)^n)). - _G. C. Greubel_, Dec 18 2021
%t Table[-(1/2)*Boole[n==0] + (1/4)*6^(n/2)*(n+1)*(3*(1+(-1)^n) + Sqrt[6]*(1-(-1)^n)), {n, 0, 40}] (* _G. C. Greubel_, Dec 18 2021 *)
%o (Magma) I:= [6,27,72,270]; [1] cat [n le 4 select I[n] else 12*(Self(n-2) - 3*Self(n-4)): n in [1..41]]; // _G. C. Greubel_, Dec 18 2021
%o (Sage)
%o @CachedFunction
%o def T(n,k): # T = A026552
%o if (k==0 or k==2*n): return 1
%o elif (k==1 or k==2*n-1): return (n+2)//2
%o elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o else: return T(n-1, k) + T(n-1, k-2)
%o @CachedFunction
%o def a(n): return sum( (k+1)*T(n,k) for k in (0..2*n) )
%o [a(n) for n in (0..40)] # _G. C. Greubel_, Dec 18 2021
%o (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -36,0,12,0]^n*[1;6;27;72])[1,1] \\ _Charles R Greathouse IV_, Oct 21 2022
%Y Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027273, A027274, A027275.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_
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