A008667 Expansion of g.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).

1, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 14, 17, 18, 22, 23, 28, 29, 34, 36, 42, 44, 50, 53, 60, 63, 71, 74, 83, 87, 96, 101, 111, 116, 127, 133, 145, 151, 164, 171, 185, 193, 207, 216, 232, 241, 258, 268, 286, 297, 316, 328, 348, 361, 382, 396, 419, 433, 457

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

Comments

Also, Molien series for invariants of finite Coxeter group A_4. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k not the alternating group Alt_k. - N. J. A. Sloane, Jan 11 2016

Number of partitions into parts 2, 3, 4, and 5. - Joerg Arndt, Apr 29 2014

Examples

			a(4)=2 because f''''(x)/4!=2 at x=0 for f=1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)).
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + 7*x^10 + 7*x^11 + ... .
		

References

• J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
• L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 32).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 241
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,-1,-2,-1,0,1,1,1,0,-1).

Crossrefs

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

Cf. A005044, A001401 (partial sums).

Programs

Magma

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) )); // G. C. Greubel, Sep 08 2019

Maple

seq(coeff(series(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)), x, n+1), x, n), n = 0..65); # G. C. Greubel, Sep 08 2019

Mathematica

SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)),{x,0,#}]&/@Range[0,100] (* or *) a[k_]=SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4) (1-x^5)),{x,0,k}] (* Peter Pein (petsie(AT)dordos.net), Sep 09 2006 *)
CoefficientList[Series[1/Times@@Table[(1-x^n),{n,2,5}],{x,0,70}],x] (* Harvey P. Dale, Feb 22 2018 *)

PARI

{a(n) = if( n<-13, -a(-14 - n), polcoeff( prod( k=2, 5, 1 / (1 - x^k), 1 + x * O(x^n)), n))} /* Michael Somos, Oct 14 2006 */

Sage

def A008667_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5))).list()
A008667_list(65) # G. C. Greubel, Sep 08 2019

Formula

Euler transform of length 5 sequence [ 0, 1, 1, 1, 1]. - Michael Somos, Sep 23 2006

a(-14 - n) = -a(n). - Michael Somos, Sep 23 2006

a(n) ~ 1/720*n^3. - Ralf Stephan, Apr 29 2014

a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6) - 2*a(n-7) - a(n-8) + a(n-10) + a(n-11) + a(n-12) - a(n-14). - David Neil McGrath, Sep 13 2014

From R. J. Mathar, Jun 23 2021: (Start)

a(n)-a(n-2) = A008680(n).

a(n)-a(n-3) = A025802(n).

a(n)-a(n-4) = A025795(n).

a(n)-a(n-5) = A005044(n+3). (End)

a(n)= floor((n^3 + 21*n^2 + 156*n - 45*n*(n mod 2) + 720)/720 - [(n mod 10)=1]/5). - Hoang Xuan Thanh, Aug 20 2025