A052934 Expansion of (1-x)/(1-6*x).

1, 5, 30, 180, 1080, 6480, 38880, 233280, 1399680, 8398080, 50388480, 302330880, 1813985280, 10883911680, 65303470080, 391820820480, 2350924922880, 14105549537280, 84633297223680, 507799783342080

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

With formula a(n) = (5*6^n + 0^n)/6, this is the binomial transform of A083425. - Paul Barry, Apr 30 2003

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

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

a(n) is the number of compositions of n when there are 5 types of each natural number. - Milan Janjic, Aug 13 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 922
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index entries for linear recurrences with constant coefficients, signature (6).

Crossrefs

Cf. A083425.

Programs

GAP

Concatenation([1], List([1..30], n-> 5*6^(n-1) )); # G. C. Greubel, Oct 18 2019

Magma

[1] cat [5*6^(n-1): n in [1..30]]; // G. C. Greubel, Oct 18 2019

Magma

R:=PowerSeriesRing(Integers(), 22); Coefficients(R!( (1-x)/(1-6*x))); // Marius A. Burtea, Oct 18 2019

Maple

spec := [S,{S=Sequence(Prod(Sequence(Z),Union(Z,Z,Z,Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(`if`(n=0,1,5*6^(n-1)), n=0..30); # G. C. Greubel, Oct 18 2019

Mathematica

Join[{1},NestList[6#&,5,20]] (* Harvey P. Dale, Nov 30 2015 *)

PARI

vector(31, n, if(n==1,1, 5*6^(n-2))) \\ G. C. Greubel, Oct 18 2019

Sage

[1]+[5*6^(n-1) for n in (1..30)] # G. C. Greubel, Oct 18 2019

Formula

a(n) = 6*a(n-1), n>=2.

a(n) = 5*6^(n-1), n>=1. - Vincenzo Librandi, Sep 15 2011

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

G.f.: 1/(1 - 5*Sum_{k>=1} x^k).

E.g.f.: (1/6)*(1 + 5*exp(6*x)). - Stefano Spezia, Oct 18 2019