A066810 Expansion of x^2/((1-3*x)*(1-2*x)^2).

0, 0, 1, 7, 33, 131, 473, 1611, 5281, 16867, 52905, 163835, 502769, 1532883, 4651897, 14070379, 42456897, 127894979, 384799049, 1156756443, 3475250065, 10436235955, 31330727961, 94038321227, 282211432673, 846835624611, 2540926304233, 7623651327931, 22872765923121

Offset: 0

Author

N. J. A. Sloane, Jan 25 2002

Keywords

• nonn
• easy

Comments

Binomial transform of A000295.

a(n) = A112626(n, 2). - Ross La Haye, Jan 11 2006

Let Q be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all x,y of P(A), xQy if x is a proper subset of y and |y| - |x| > 1. Then a(n) = |Q|. - Ross La Haye, Jan 11 2008

a(n) is the number of n-digit ternary sequences that have at least two 0's. - Geoffrey Critzer, Apr 14 2009

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
• Index entries for linear recurrences with constant coefficients, signature (7,-16,12).

Crossrefs

Column k=1 of A238858 (with different offset).

Programs

GAP

List([0..30], n-> 3^n - 2^n - n*2^(n-1)); # G. C. Greubel, Nov 18 2019

Magma

[3^n-2^n-n*2^(n-1): n in [0..30]]; // Vincenzo Librandi, Nov 29 2015

Maple

seq(3^n - 2^n - n*2^(n-1), n=0..30); # G. C. Greubel, Nov 18 2019

Mathematica

RecurrenceTable[{a[n]==3*a[n-1] + (n-1) 2^(n-2), a[0]==0}, a, {n, 0, 30}] (* Geoffrey Critzer, Apr 14 2009 *)
CoefficientList[Series[x^2/((1-3x)(1-2x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 29 2015 *)

PARI

a(n) = 3^n -2^n -n*2^(n-1) \\ Harry J. Smith, Mar 29 2010

Sage

[3^n - 2^n - n*2^(n-1) for n in (0..30)] # G. C. Greubel, Nov 18 2019

Formula

a(n) = 3^n - 2^n - n*2^(n-1).

From Ross La Haye, Apr 26 2006: (Start)

a(n) = A000244(n) - A001792(n).

a(n) = Sum_{k=2..n} binomial(n,k)2^(n-k). (End)

Inverse binomial transform of A086443. - Ross La Haye, Apr 29 2006

Convolution of A000244 beginning [0,1,3,9,27,81,...] and A001787. - Ross La Haye, Feb 15 2007

From Geoffrey Critzer, Apr 14 2009: (Start)

E.g.f.: exp(2*x)*(exp(x) - x - 1).

a(n) = 3*a(n-1) + (n-1)*2^(n-2). (End)

Extensions

Additional comments from Ross La Haye, Sep 27 2005