A113071 Expansion of g.f. ((1+x)/(1-3*x))^2.

1, 8, 40, 168, 648, 2376, 8424, 29160, 99144, 332424, 1102248, 3621672, 11809800, 38263752, 123294312, 395392104, 1262703816, 4017693960, 12741829416, 40291730856, 127073920392, 399817944648, 1255242384360, 3933092804328

Offset: 0

Author

Paul Barry, Oct 14 2005

Keywords

• easy
• nonn

Comments

Binomial transform is A014916. 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-9).

Crossrefs

Cf. A003946, A014916, A081038.

Programs

GAP

a:=[1,8,40];; for n in [4..30] do a[n]:=6*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, May 24 2019

Magma

I:=[8,40]; [1] cat [n le 2 select I[n] else 6*Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, May 24 2019

Mathematica

CoefficientList[Series[(1+x)^2/(1-3x)^2, {x, 0, 30}], x] (* Georg Fischer, May 24 2019 *)
LinearRecurrence[{6,-9}, {1,8,40}, 30] (* G. C. Greubel, May 24 2019 *)

PARI

my(x='x+O('x^30)); Vec(((1+x)/(1-3*x))^2) \\ G. C. Greubel, May 24 2019

Sage

(((1+x)/(1-3*x))^2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

Formula

G.f.: (1+x)^2/(1-3*x)^2. [Corrected by Georg Fischer, May 24 2019]

a(n) = (8*3^n*(2*n+1) + 0^n)/9 = (4*3^n*(4*n+2) + 0^n)/9;

a(n) = Sum_{k=0..n} A003946(k)*A003946(n-k).

a(n) = Sum_{k=0..n} C(n, k)*Sum_{j=0..k} (j+1)*4^j.

a(n) = 8*A081038(n-1), n>0. - R. J. Mathar, Nov 28 2014

E.g.f.: (1 + 8*exp(3*x)*(1 + 6*x))/9. - Stefano Spezia, Jan 31 2025