A082149 A transform of C(n,2).

0, 0, 1, 6, 30, 140, 615, 2562, 10220, 39384, 147645, 541310, 1948650, 6908772, 24180611, 83702010, 286978200, 975725744, 3293074233, 11041484022, 36804946550, 122037454140, 402723598431, 1323234680306, 4330586226180

Offset: 0

Author

Paul Barry, Apr 07 2003

Keywords

• easy
• nonn

Comments

Represents the mean of C(n,2) with its second binomial transform. Binomial transform of A080929 (preceded by two zeros).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-57,136,-171,108,-27).

Crossrefs

Cf. A082150, A000217, A027472.

Programs

Magma

[Binomial(n,2)*(1 + 3^(n-2))/2: n in [0..30]]; // G. C. Greubel, Feb 10 2018

Mathematica

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 *)
LinearRecurrence[{12,-57,136,-171,108,-27},{0,0,1,6,30,140},30] (* Harvey P. Dale, Aug 11 2021 *)

PARI

for(n=0,30, print1(binomial(n,2)*(1 + 3^(n-2))/2, ", ")) \\ G. C. Greubel, Feb 10 2018

Formula

a(n) = C(n, 2)*(3^(n-2) + 1)/2.

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

G.f.: x^2(14*x^3-15*x^2+6*x-1)/((1-x)^3*(3*x-1)^3).

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