A016129 Expansion of 1/((1-2*x)*(1-6*x)).

1, 8, 52, 320, 1936, 11648, 69952, 419840, 2519296, 15116288, 90698752, 544194560, 3265171456, 19591036928, 117546237952, 705277460480, 4231664828416, 25389989101568, 152339934871552, 914039609753600, 5484237659570176, 32905425959518208, 197432555761303552

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..900
• Index entries for linear recurrences with constant coefficients, signature (8,-12).

Crossrefs

Row sums of A100851.

Cf. A071951, A089278, A089500, A129467, A144843.

Sequences with gf 1/((1-n*x)*(1-6*x)): A000400 (n=0), A003464 (n=1), this sequence (n=2), A016137 (n=3), A016149 (n=4), A005062 (n=5), A053469 (n=6), A016169 (n=7), A016170 (n=8), A016172 (n=9), A016173 (n=10), A016174 (n=11), A016175 (n=12).

Programs

Magma

[(6^(n+1)-2^(n+1))/4 : n in [0..30]]; // Vincenzo Librandi, Oct 09 2011

Mathematica

Table[(6^(n+1) -2^(n+1))/4, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[1/((1-2x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-12},{1,8},30] (* Harvey P. Dale, Jan 15 2015 *)

PARI

Vec(1/(1-2*x)/(1-6*x)+O(x^30)) \\ Charles R Greathouse IV, Apr 17 2012

Sage

[lucas_number1(n,8,12) for n in range(1, 31)] # Zerinvary Lajos, Apr 23 2009

Sage

[(6^n - 2^n)/4 for n in range(1,31)] # Zerinvary Lajos, Jun 04 2009

Formula

a(n) = A071951(n+2, 2) = 9*(2*3)^(n-1) - (2*1)^(n-1) = (2^(n-1))*(3^(n+1)-1), n>=0. - Wolfdieter Lang, Nov 07 2003

From Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 05 2005: (Start)

G.f.: 1/((1-2*x)*(1-6*x)).

E.g.f.: (-exp(2*x) + 3*exp(6*x))/2.

a(n) = (6^(n+1) - 2^(n+1))/4. (End)

a(n)^2 = A144843(n+1). - Philippe Deléham, Nov 26 2008

a(n) = 8*a(n-1) - 12*a(n-2). - Philippe Deléham, Jan 01 2009

a(n) = det(|ps(i+2,j+1)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - Mircea Merca, Apr 06 2013