A113954 Expansion of (1-2x^2)/((1-2x)(1+x)^2).

1, 0, 1, 2, 3, 8, 13, 30, 55, 116, 225, 458, 907, 1824, 3637, 7286, 14559, 29132, 58249, 116514, 233011, 466040, 932061, 1864142, 3728263, 7456548, 14913073, 29826170, 59652315, 119304656, 238609285, 477218598, 954437167, 1908874364, 3817748697

Offset: 0

Author

Paul Barry, Nov 09 2005

Keywords

• easy
• nonn

Comments

Inverse binomial transform of phi(phi(3^n)).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,2).

Crossrefs

Cf. A103196.

Programs

Mathematica

CoefficientList[Series[(1-2x^2)/((1-2x)(1+x)^2),{x,0,40}],x] (* or *) LinearRecurrence[{0,3,2},{1,0,1},40] (* Harvey P. Dale, Aug 20 2015 *)

Formula

a(n)=3a(n-2)+2a(n-3); a(n)=2^(n+1)/9+(7-3n)(-1)^n/9; a(n)=a(n)=sum{k=0..n, (-1)^(n-k)*C(n, k)phi(phi(3^k))}; a(n)=sum{k=0..n, (-1)^(n-k)*C(n, k)(2*3^k/9+C(1, k)/3+4*C(0, k)/9)}; a(n)=sum{k=0..n, J(n-k+1)((-1)^(k+1)-2C(1, k)+4C(0, k))} where J(n)=A001045(n).