A008727 Molien series for 3-dimensional group [2,n] = *22n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of partitions of n into two kinds of 1's and one kind of 9. - Joerg Arndt, Dec 27 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 192
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,1,-2,1).

Crossrefs

Cf. A001840, A001972, A008724, A008725, A008726, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010

Programs

GAP

a:=[1,2,3,4,5,6,7,8,9,11,13];; for n in [12..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-9]-2*a[n-10]+a[n-11]; od; a; # G. C. Greubel, Sep 09 2019

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^9)) )); // G. C. Greubel, Sep 09 2019

Maple

seq(coeff(series(1/((1-x)^2*(1-x^9)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 09 2019

Mathematica

Drop[Accumulate[Floor[Range[70]/9]], 8] (* Jean-François Alcover, Mar 27 2013 *)
CoefficientList[Series[1/(1-x)^2/(1-x^9), {x,0,70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,11,13},120] (* Harvey P. Dale, Feb 13 2022 *)

PARI

Vec(1/(1-x)^2/(1-x^9)+O(x^66)) /* Joerg Arndt, Mar 27 2013 */

Sage

def A008727_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)^2*(1-x^9))).list()
A008727_list(70) # G. C. Greubel, Sep 09 2019

Formula

G.f.: 1/((1-x)^2*(1-x^9)).

From Mitch Harris, Sep 08 2008: (Start)

a(n) = Sum_{j=0..n+9} floor(j/9).

a(n-9) = (1/2)*floor(n/9)*(2*n - 7 - 9*floor(n/9)). (End)