A201458 Expansion of 1/((1-2*x)*(1-3*x+3*x^2)*(1-4*x+6*x^2-4*x^3)).

1, 9, 46, 175, 551, 1520, 3811, 8921, 19922, 43211, 92363, 196608, 419295, 897565, 1926458, 4135255, 8854359, 18875392, 40024059, 84417521, 177221602, 370688979, 773342163, 1610612736, 3350668423, 6964989333, 14466833194, 30021724351, 62233946303

Offset: 0

Author

Bruno Berselli, Dec 01 2011

Keywords

• nonn
• easy

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-35,76,-98,72,-24).

Crossrefs

Cf. for type of g.f.: A099855, with 1/((1-2*x+2*x^2)*(1-4*x+4*x^2)); A000581, with 1/((1-x)^2*(1-x)^3*(1-x)^4).

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-2*x)*(1-3*x+3*x^2)*(1-4*x+6*x^2-4*x^3))));

Mathematica

CoefficientList[Series[1/((1-2*x)*(1-3*x+3*x^2)*(1-4*x+6*x^2-4*x^3)), {x, 0, 30}], x]
LinearRecurrence[{9,-35,76,-98,72,-24},{1,9,46,175,551,1520},30] (* Harvey P. Dale, Feb 01 2012 *)

Maxima

makelist(coeff(taylor(1/((1-2*x)*(1-3*x+3*x^2)*(1-4*x+6*x^2-4*x^3)), x, 0, n), x, n), n, 0, 29);

PARI

Vec(1/((1-2*x)*(1-3*x+3*x^2)*(1-4*x+6*x^2-4*x^3))+O(x^30))

Formula

G.f.: 1/((1-2*x)^2*(1-2*x+2*x^2)*(1-3*x+3*x^2)) = 1/((1-2*x+2*x^2)*(1-3*x+3*x^2)*(1-4*x+4*x^2)).

a(n) = 9*a(n-1)-35*a(n-2)+76*a(n-3)-98*a(n-4)+72*a(n-5)-24*a(n-6) for a(-5)=a(-4)=a(-3)=a(-2)=a(-1)=0, a(0)=1.

a(n) = 8*2^n*(n+1)+2*((1-i)^(n-1)+(1+i)^(n-1))+((3+i*sqrt(3))/2)^(n+4)+((3-i*sqrt(3))/2)^(n+4), where i=sqrt(-1).