|
%I #55 Feb 05 2022 16:36:13
%S 1,5,30,180,1080,6480,38880,233280,1399680,8398080,50388480,302330880,
%T 1813985280,10883911680,65303470080,391820820480,2350924922880,
%U 14105549537280,84633297223680,507799783342080
%N Expansion of (1-x)/(1-6*x).
%C With formula a(n) = (5*6^n + 0^n)/6, this is the binomial transform of A083425. - _Paul Barry_, Apr 30 2003
%C For n>=1, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5,6} we have f(x) != y. - Aleksandar M. Janjic and _Milan Janjic_, Mar 27 2007
%C a(n) = (n+1) terms in the sequence (1, 4, 5, 5, 5, ...) dot (n+1) terms in the sequence (1, 1, 5, 30, 180, 1080, ...). Example: a(4) = (1, 4, 5, 5, 5) dot (1, 1, 5, 30, 180) = (1 + 4 + 25 + 150 + 900), where (1, 4, 25, 150, ...) = first differences of current sequence. - _Gary W. Adamson_, Aug 03 2010
%C a(n) is the number of compositions of n when there are 5 types of each natural number. - _Milan Janjic_, Aug 13 2010
%H Vincenzo Librandi, <a href="/A052934/b052934.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=922">Encyclopedia of Combinatorial Structures 922</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (6).
%F a(n) = 6*a(n-1), n>=2.
%F a(n) = 5*6^(n-1), n>=1. - _Vincenzo Librandi_, Sep 15 2011
%F G.f.: (1-x)/(1-6*x).
%F G.f.: 1/(1 - 5*Sum_{k>=1} x^k).
%F E.g.f.: (1/6)*(1 + 5*exp(6*x)). - _Stefano Spezia_, Oct 18 2019
%p spec := [S,{S=Sequence(Prod(Sequence(Z),Union(Z,Z,Z,Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%p seq(`if`(n=0,1,5*6^(n-1)), n=0..30); # _G. C. Greubel_, Oct 18 2019
%t Join[{1},NestList[6#&,5,20]] (* _Harvey P. Dale_, Nov 30 2015 *)
%o (PARI) vector(31, n, if(n==1,1, 5*6^(n-2))) \\ _G. C. Greubel_, Oct 18 2019
%o (Magma) [1] cat [5*6^(n-1): n in [1..30]]; // _G. C. Greubel_, Oct 18 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 22); Coefficients(R!( (1-x)/(1-6*x))); // _Marius A. Burtea_, Oct 18 2019
%o (Sage) [1]+[5*6^(n-1) for n in (1..30)] # _G. C. Greubel_, Oct 18 2019
%o (GAP) Concatenation([1], List([1..30], n-> 5*6^(n-1) )); # _G. C. Greubel_, Oct 18 2019
%Y Cf. A083425.
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
|
