0,2

Binomial transform is A014915. In general, ((1+x)/(1-r*x))^2 expands to a(n) = ((r+1)*r^n*((r+1)*n + r - 1) + 0^n)/r^2, which is also a(n) = Sum_{k=0..n} C(n,k)*Sum_{j=0..k} (j+1)*(r+1)^j. This is the self-convolution of the coordination sequence for the infinite tree with valency r.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-4).

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

a(n) = 3*2^n(3n+1)/4 + 0^n/4;

a(n) = Sum_{k=0..n} A003945(k)*A003945(n-k);

a(n) = Sum_{k=0..n} C(n, k)*Sum_{j=0..k} (j+1)*3^j.

a(n) = 4*a(n-1) - 4*a(n-2); a(0)=1, a(1)=6, a(2)=21. - Harvey P. Dale, May 20 2011

Sum_{n>=0} (-1)^n/a(n) = (2^(4/3)*(log(1+2^(1/3)+2^(2/3)) + 2*sqrt(3)*arccot((2^(4/3)-1)/sqrt(3))) - 3)/9. - Amiram Eldar, Feb 04 2026

Join[{1}, LinearRecurrence[{4, -4}, {6, 21}, 30]] (* or *) CoefficientList[ Series[((1+x)/(1-2x))^2, {x, 0, 30}], x] (* Harvey P. Dale, May 20 2011 *)

(Magma) [3*2^n*(3*n+1)/4+0^n/4: n in [0..30]]; // Vincenzo Librandi, May 21 2011

Cf. A003945, A014915, A113071.

Sequence in context: A321257 A305120 A066524 * A009147 A012593 A276235

Adjacent sequences: A113067 A113068 A113069 * A113071 A113072 A113073

easy,nonn

Paul Barry, Oct 14 2005

approved