A169792 Expansion of ((1-x)/(1-2x))^5.

1, 5, 20, 70, 225, 681, 1970, 5500, 14920, 39520, 102592, 261760, 657920, 1632000, 4001280, 9708544, 23336960, 55623680, 131563520, 309002240, 721092608, 1672806400, 3859415040, 8859156480, 20240138240, 46038777856, 104291368960, 235342397440, 529153392640

Offset: 0

Author

N. J. A. Sloane, May 15 2010

Keywords

• nonn
• easy

Comments

a(n) is the number of weak compositions of n with exactly 4 parts equal to 0. - Milan Janjic, Jun 27 2010

Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^5; see A291000. - Clark Kimberling, Aug 24 2017

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..500
• Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
• Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
• M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
• Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Crossrefs

((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

Programs

GAP

Concatenation([1],List([1..30],n->2^n*(n+4)*(n^3+26*n^2+171*n+186)/768)); # Muniru A Asiru, Aug 22 2018

Maple

seq(coeff(series(((1-x)/(1-2*x))^5, x,n+1),x,n),n=0..30); # Muniru A Asiru, Aug 22 2018

Mathematica

CoefficientList[Series[((1 - x)/(1 - 2 x))^5, {x, 0, 28}], x] (* Michael De Vlieger, Oct 15 2018 *)

PARI

Vec(((1-x)/(1-2*x))^5+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Formula

a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5), n >= 6. - Vincenzo Librandi, Mar 14 2011

a(n) = 2^n*(n+4)*(n^3 + 26*n^2 + 171*n + 186)/768, n > 0. - R. J. Mathar, Mar 14 2011