A082151 - OEIS

%I #13 Apr 10 2023 10:40:11

%S 0,0,1,12,102,760,5295,35364,228956,1445616,8936685,54252220,

%T 324214242,1911205608,11132579003,64170616020,366497915640,

%U 2076171038176,11676266706969,65242364726124,362433045180830,2002838101907160,11015341078090503,60321223747375492

%N A transform of C(n,2).

%C Represents the mean of the second and fourth binomial transforms of C(n,2). Binomial transform of A082150

%H G. C. Greubel, <a href="/A082151/b082151.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 (24,-237,1232,-3555,5400,-3375).

%F a(n) = C(n, 2)*(3^(n-2) + 5^(n-2))/2.

%F G.f.: (x^2/(1-5*x)^3 + x^2/(1-3*x)^3)/2.

%F a(n) = x^2*(76*x^3 - 51*x^2 + 12*x - 1)/((1-3*x)^3*(5*x-1)^3).

%F E.g.f.: x^2*exp(4*x)*cosh(x)/2.

%t CoefficientList[Series[(x^2/(1-5*x)^3 + x^2/(1-3*x)^3)/2, {x,0,50}], x] (* or *) Table[Binomial[n,2]*(3^(n-2) + 5^(n-2))/2, {n,0,30}] (* _G. C. Greubel_, Feb 10 2018 *)

%t LinearRecurrence[{24,-237,1232,-3555,5400,-3375},{0,0,1,12,102,760},30] (* _Harvey P. Dale_, Apr 10 2023 *)

%o (PARI) for(n=0,30, print1(binomial(n,2)*(3^(n-2) + 5^(n-2))/2, ", ")) \\ _G. C. Greubel_, Feb 10 2018

%o (Magma) [Binomial(n,2)*(3^(n-2) + 5^(n-2))/2: n in [0..30]]; // _G. C. Greubel_, Feb 10 2018

%Y Cf. A027472, A081135, A000217.

%K easy,nonn

%O 0,4

%A _Paul Barry_, Apr 07 2003