%I #55 Apr 24 2020 21:01:04
%S 1,5,20,70,225,681,1970,5500,14920,39520,102592,261760,657920,1632000,
%T 4001280,9708544,23336960,55623680,131563520,309002240,721092608,
%U 1672806400,3859415040,8859156480,20240138240,46038777856,104291368960,235342397440,529153392640
%N Expansion of ((1-x)/(1-2x))^5.
%C a(n) is the number of weak compositions of n with exactly 4 parts equal to 0. - _Milan Janjic_, Jun 27 2010
%C 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
%H Muniru A Asiru, <a href="/A169792/b169792.txt">Table of n, a(n) for n = 0..500</a>
%H Robert Davis, Greg Simay, <a href="https://arxiv.org/abs/2001.11089">Further Combinatorics and Applications of Two-Toned Tilings</a>, arXiv:2001.11089 [math.CO], 2020.
%H Nickolas Hein, Jia Huang, <a href="https://arxiv.org/abs/1807.04623">Variations of the Catalan numbers from some nonassociative binary operations</a>, arXiv:1807.04623 [math.CO], 2018.
%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H M. Janjic, B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32).
%F 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
%F a(n) = 2^n*(n+4)*(n^3 + 26*n^2 + 171*n + 186)/768, n > 0. - _R. J. Mathar_, Mar 14 2011
%p seq(coeff(series(((1-x)/(1-2*x))^5, x,n+1),x,n),n=0..30); # _Muniru A Asiru_, Aug 22 2018
%t CoefficientList[Series[((1 - x)/(1 - 2 x))^5, {x, 0, 28}], x] (* _Michael De Vlieger_, Oct 15 2018 *)
%o (PARI) Vec(((1-x)/(1-2*x))^5+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o (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
%Y ((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, May 15 2010