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%I #16 Sep 08 2022 08:45:09
%S 0,0,1,6,30,140,615,2562,10220,39384,147645,541310,1948650,6908772,
%T 24180611,83702010,286978200,975725744,3293074233,11041484022,
%U 36804946550,122037454140,402723598431,1323234680306,4330586226180
%N A transform of C(n,2).
%C Represents the mean of C(n,2) with its second binomial transform. Binomial transform of A080929 (preceded by two zeros).
%H G. C. Greubel, <a href="/A082149/b082149.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 (12,-57,136,-171,108,-27).
%F a(n) = C(n, 2)*(3^(n-2) + 1)/2.
%F G.f.: (x^2/(1-3x)^3+x^2/(1-x)^3)/2.
%F G.f.: x^2(14*x^3-15*x^2+6*x-1)/((1-x)^3*(3*x-1)^3).
%F E.g.f.: x^2*exp(2*x)*cosh(x)/2.
%t CoefficientList[Series[(x^2/(1-3*x)^3 + x^2/(1-x)^3)/2, {x,0,50}], x] (* or *) Table[Binomial[n,2]*(1 + 3^(n-2))/2, {n,0,30}] (* _G. C. Greubel_, Feb 10 2018 *)
%t LinearRecurrence[{12,-57,136,-171,108,-27},{0,0,1,6,30,140},30] (* _Harvey P. Dale_, Aug 11 2021 *)
%o (PARI) for(n=0,30, print1(binomial(n,2)*(1 + 3^(n-2))/2, ", ")) \\ _G. C. Greubel_, Feb 10 2018
%o (Magma) [Binomial(n,2)*(1 + 3^(n-2))/2: n in [0..30]]; // _G. C. Greubel_, Feb 10 2018
%Y Cf. A082150, A000217, A027472.
%K easy,nonn
%O 0,4
%A _Paul Barry_, Apr 07 2003
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