%I #22 Oct 18 2022 14:52:38
%S 1,6,18,72,180,648,1512,5184,11664,38880,85536,279936,606528,1959552,
%T 4199040,13436928,28553472,90699264,191476224,604661760,1269789696,
%U 3990767616,8344332288,26121388032,54419558400,169789022208
%N a(n) = Sum_{k=0..2n} (k+1) * A026519(n, k).
%H G. C. Greubel, <a href="/A027266/b027266.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) * A026519(n, k).
%F G.f.: (1+6*x+6*x^2)/(1-6*x^2)^2.
%F a(n) = 12*a(n-2) - 36*a(n-4), with a(0)=1, a(1)=6, a(2)=18, a(3)=72. - _Harvey P. Dale_, Jun 19 2015
%F a(n) = ((n+1)/2)*6^((n-1)/2)*( 3*(1-(-1)^n) + sqrt(6)*(1+(-1)^n) ). - _G. C. Greubel_, Dec 21 2021
%t CoefficientList[Series[(1+6x+6x^2)/(1-6x^2)^2,{x,0,30}],x] (* or *) LinearRecurrence[{0,12,0,-36},{1,6,18,72},30] (* _Harvey P. Dale_, Jun 19 2015 *)
%o (Magma) I:=[1,6,18,72]; [n le 4 select I[n] else 12*(Self(n-2) - 3*Self(n-4)): n in [1..41]]; // _G. C. Greubel_, Dec 21 2021
%o (Sage) [((n+1)/2)*6^((n-1)/2)*( 3*(1-(-1)^n) + sqrt(6)*(1+(-1)^n) ) for n in (0..40)] # _G. C. Greubel_, Dec 21 2021
%o (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -36,0,12,0]^n*[1;6;18;72])[1,1] \\ _Charles R Greathouse IV_, Oct 18 2022
%Y Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026532, A026533, A026534, A027262, A027263, A027264, A027265.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_