A008720 Molien series for 3-dimensional group [2,5] = *225.

1, 0, 2, 0, 3, 1, 4, 2, 5, 3, 7, 4, 9, 5, 11, 7, 13, 9, 15, 11, 18, 13, 21, 15, 24, 18, 27, 21, 30, 24, 34, 27, 38, 30, 42, 34, 46, 38, 50, 42, 55, 46, 60, 50, 65, 55, 70, 60, 75, 65, 81, 70, 87, 75, 93, 81, 99, 87, 105, 93, 112, 99, 119, 105, 126, 112, 133, 119, 140, 126, 148, 133, 156

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of partitions of n into parts 2 and 5 where there are two kinds of parts 2. - Hoang Xuan Thanh, Jun 20 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 223
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1,1,0,-2,0,1).

Programs

GAP

a:=[1,0,2,0,3,1,4,2,5];; for n in [10..80] do a[n]:=2*a[n-2]-a[n-4] +a[n-5]-2*a[n-7]+a[n-9]; od; a; # G. C. Greubel, Sep 09 2019

Magma

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^2)^2*(1-x^5)) )); // G. C. Greubel, Sep 09 2019

Maple

1/((1-x^2)^2*(1-x^5)); seq(coeff(series(%, x, n+1), x, n), n = 0 .. 80); # modified by G. C. Greubel, Sep 09 2019

Mathematica

LinearRecurrence[{0,2,0,-1,1,0,-2,0,1}, {1,0,2,0,3,1,4,2,5}, 80] (* Harvey P. Dale, Dec 10 2015 *)

PARI

my(x='x+O('x^80)); Vec(1/((1-x^2)^2*(1-x^5))) \\ G. C. Greubel, Sep 09 2019

Sage

def A008720_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^2)^2*(1-x^5))).list()
A008720_list(80) # G. C. Greubel, Sep 09 2019

Formula

a(n) = floor((n^2 + n*(9+5*(-1)^n) + 23*(-1)^n + 26)/40). - Hoang Xuan Thanh, Jun 20 2025

Extensions

Terms a(65) onward added by G. C. Greubel, Sep 09 2019