%I #24 Sep 08 2022 08:45:09
%S 0,0,1,9,60,360,2040,11088,58240,297216,1480320,7223040,34636800,
%T 163657728,763549696,3523645440,16107110400,73016672256,328570011648,
%U 1468890021888,6528375193600,28862235279360,126993714118656
%N A transform of C(n,2).
%C Represents the mean of the first and third binomial transforms of C(n,2) Binomial transform of A082149.
%H G. C. Greubel, <a href="/A082150/b082150.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (18,-132,504,-1056,1152,-512).
%F a(n) = C(n, 2)*(2^(n-2) + 4^(n-2))/2.
%F G.f.: (x^2/(1-2*x)^3 + x^2/(1-4*x)^3)/2.
%F G.f.: x^2*(36*x^3 - 30*x^2 + 9*x-1)/((1 - 2*x)^3*(4*x - 1)^3).
%F E.g.f.: x^2*exp(3*x)*cosh(x)/2.
%F From _Bruno Berselli_, Feb 12 2018: (Start)
%F E.g.f.: x^2*(1 + exp(2*x))*exp(2*x)/4.
%F a(n) = 2^(n-4)*(2^(n-2) + 1)*(n - 1)*n. (End)
%p A082150:=[seq(binomial(n,2)*(2^(n-2)+4^(n-2))/2,n=0..23)]; # _Muniru A Asiru_, Feb 12 2018
%t CoefficientList[Series[(x^2/(1-2*x)^3 + x^2/(1-4*x)^3)/2, {x,0,50}], x] (* or *) Table[Binomial[n,2]*(2^(n-2) + 4^(n-2))/2, {n,0,30}] (* _G. C. Greubel_, Feb 10 2018 *)
%t LinearRecurrence[{18,-132,504,-1056,1152,-512},{0,0,1,9,60,360},30] (* _Harvey P. Dale_, Jan 17 2022 *)
%o (PARI) for(n=0,30, print1(binomial(n,2)*(2^(n-2) + 4^(n-2))/2, ", ")) \\ _G. C. Greubel_, Feb 10 2018
%o (Magma) [Binomial(n,2)*(2^(n-2) + 4^(n-2))/2: n in [0..30]]; // _G. C. Greubel_, Feb 10 2018
%o (GAP) List([0..23], n-> Binomial(n,2)*(2^(n-2)+4^(n-2))/2); # _Muniru A Asiru_, Feb 12 2018
%o (Maxima) makelist(2^(n-4)*(2^(n-2)+1)*(n-1)*n, n, 0, 30); /* _Bruno Berselli_, Feb 13 2018 */
%Y Cf. A082151, A038845, A001788, A000217.
%K nonn,easy
%O 0,4
%A _Paul Barry_, Apr 07 2003