A169793 Expansion of ((1-x)/(1-2*x))^6.

1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, 236640, 632448, 1661056, 4296192, 10961664, 27630592, 68889600, 170065920, 416071680, 1009582080, 2431254528, 5814222848, 13815054336, 32629850112, 76640681984, 179080003584, 416412598272, 963876225024

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 5 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)^6; see A291000. - Clark Kimberling, Aug 24 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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 (12,-60,160,-240,192,-64).

Crossrefs

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

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^6)); // G. C. Greubel, Oct 16 2018

Maple

seq(coeff(series(((1-x)/(1-2*x))^6,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 16 2018

Mathematica

CoefficientList[Series[((1 - x)/(1 - 2 x))^6, {x, 0, 27}], x] (* Michael De Vlieger, Oct 15 2018 *)

PARI

x='x+O('x^30); Vec(((1-x)/(1-2*x))^6) \\ G. C. Greubel, Oct 16 2018

Formula

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

For n > 0, a(n) = 2^(n-9)*(n+7)*(n^4 + 38*n^3 + 419*n^2 + 1342*n + 1080)/15. - Bruno Berselli, Aug 07 2011