A082151 A transform of C(n,2).

0, 0, 1, 12, 102, 760, 5295, 35364, 228956, 1445616, 8936685, 54252220, 324214242, 1911205608, 11132579003, 64170616020, 366497915640, 2076171038176, 11676266706969, 65242364726124, 362433045180830, 2002838101907160, 11015341078090503, 60321223747375492

Offset: 0

Author

Paul Barry, Apr 07 2003

Keywords

• easy
• nonn

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (24,-237,1232,-3555,5400,-3375).

Crossrefs

Cf. A027472, A081135, A000217.

Programs

Magma

[Binomial(n,2)*(3^(n-2) + 5^(n-2))/2: n in [0..30]]; // G. C. Greubel, Feb 10 2018

Mathematica

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

PARI

for(n=0,30, print1(binomial(n,2)*(3^(n-2) + 5^(n-2))/2, ", ")) \\ G. C. Greubel, Feb 10 2018

Formula

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

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

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

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