A227209 Expansion of 1/((1-x)^2*(1-2x)*(1-4x)).

1, 8, 43, 198, 849, 3516, 14311, 57746, 231997, 930024, 3724179, 14904894, 59635945, 238576532, 954371647, 3817617642, 15270732693, 61083455040, 244334868715, 977341571990, 3909370482241, 15637490317548

Offset

0,2

Comments

This sequence was chosen to illustrate a method of solution.

In general, for the expansion of 1/((1-t)^2)*(1-s)(1-r)) with r>s>t we have the formula: a(n) = ( K*r^(n+3) + L*s^(n+3) + M*t^(n+3) + N*t^(n+3) )/D where K,L,M,N,D, have the following values:

K = (s-t)^2;

L = -(r-t)^2;

M = (r-s)*(r+s-2*t);

N = (r-t)*(s-t)*(r-s)*(n+3);

D = (r-s)*(r-t)^2*(s-t)^2.

Directly using formula we get a(n) = ( 4^(n+3) - 9*2^(n+3) + 8 + 6*(n+3) )/18. After transformation we obtain previous formula.

Links

Table of n, a(n) for n=0..21.

Index entries for linear recurrences with constant coefficients, signature (8, -21, 22, -8).

Formula

G.f.: 1/((1-x)^2*(1-2*x)*(1-4*x)).

a(n) = ( 4^(n+3) - 9*2^(n+3) + 6*n + 26 )/18.

Mathematica

nn = 25; CoefficientList[Series[1/((1 - x)^2*(1 - 2 x)*(1 - 4 x)), {x, 0, nn}], x] (* T. D. Noe, Sep 19 2013 *)

Crossrefs

Cf. A229026.

Partial sums of A171477.

Sequence in context: A094865 A122880 A171479 * A282523 A099253 A239033

Adjacent sequences: A227206 A227207 A227208 * A227210 A227211 A227212

Keyword

nonn,easy

Author

Yahia Kahloune, Sep 19 2013

Status

approved