login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A113071 Expansion of ((1+x)/(1-3*x))^2. 3
1, 8, 40, 168, 648, 2376, 8424, 29160, 99144, 332424, 1102248, 3621672, 11809800, 38263752, 123294312, 395392104, 1262703816, 4017693960, 12741829416, 40291730856, 127073920392, 399817944648, 1255242384360, 3933092804328 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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

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 *)

PROG

(PARI) my(x='x+O('x^30)); Vec(((1+x)/(1-3*x))^2) \\ 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

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

(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

CROSSREFS

Sequence in context: A001789 A074412 A217375 * A006726 A165665 A000760

Adjacent sequences:  A113068 A113069 A113070 * A113072 A113073 A113074

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Oct 14 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 18 03:55 EDT 2021. Contains 347505 sequences. (Running on oeis4.)