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%I #34 May 14 2023 09:40:18
%S 1,4,18,48,180,432,1512,3456,11664,25920,85536,186624,606528,1306368,
%T 4199040,8957952,28553472,60466176,191476224,403107840,1269789696,
%U 2660511744,8344332288,17414258688,54419558400,113192681472,352638738432,731398864896,2272560758784
%N a(n) = Sum_{k=0..2n} (k+1)*T(n,k), where T is given by A026536.
%H G. C. Greubel, <a href="/A027271/b027271.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 From _Paul Barry_, Mar 03 2004: (Start)
%F G.f.: (1+4*x+6*x^2)/(1-6*x^2)^2 = (d/dx)((1+3*x)/(1-6*x^2)).
%F a(n) = 6^(n/2)*((3-sqrt(6))*(-1)^n + (3+sqrt(6)))*(n+1)/6. (End)
%F a(n) = 4*b(n) + b(n+1) + 6*b(n-1) with b(n)= 0, 1, 0, 12, 0, 108, 0, 864, ... (aerated A053469). - _R. J. Mathar_, Sep 29 2012
%F E.g.f.: (1 + 2*x)*cosh(sqrt(6)*x) + sqrt(2/3)*(1 + 3*x)*sinh(sqrt(6)*x). - _Stefano Spezia_, May 07 2023
%t T[n_, k_]:= T[n, k]= If[k==0 || k==2*n, 1, If[k==1 || k==2*n-1, Floor[n/2], If[EvenQ[n], T[n-1, k-2] +T[n-1, k-1] +T[n-1, k], T[n-1, k-2] +T[n-1, k]] ]];
%t A027271[n_]:= A027271[n]= Sum[(k+1)*T[n,k], {k,0,2*n}];
%t Table[A027271[n], {n,0,40}] (* _G. C. Greubel_, Apr 12 2022 *)
%o (PARI) A027271(n)=my(b(n)=if(!bittest(n,0),n\2*6^(n\2-1)));4*b(n+1)+b(n+2)+6*b(n) \\ could be made more efficient and explicit by simplifying the formula for n even and for n odd separately. - _M. F. Hasler_, Sep 29 2012
%o (Magma) [Round(6^(n/2)*( 3*((n+1) mod 2) + Sqrt(6)*(n mod 2) )*(n+1)/3): n in [0..40]]; // _G. C. Greubel_, Apr 12 2022
%o (SageMath) [6^(n/2)*( 3*((n+1)%2) + sqrt(6)*(n%2) )*(n+1)/3 for n in (0..40)] # _G. C. Greubel_, Apr 12 2022
%Y Cf. A026536, A053469, A199299 (bisection).
%K nonn,easy
%O 0,2
%A _Clark Kimberling_
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