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

0,5

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

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).

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

a(n)-a(n-2) = A008680(n). - R. J. Mathar, Jun 23 2021

a(n)-a(n-3) = A025802(n). - R. J. Mathar, Jun 23 2021

a(n)-a(n-4) = A025795(n). - R. J. Mathar, Jun 23 2021

a(n)-a(n-5) = A005044(n+3). - R. J. Mathar, Jun 23 2021

Example

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 + ... .

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 *)

Prog

(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 */
(Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) )); // G. C. Greubel, Sep 08 2019
(Sage)
def A008667_list(prec):
    P.<x> = 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

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).

Sequence in context: A349675 A029019 A040039 * A367221 A239880 A240862

Adjacent sequences: A008664 A008665 A008666 * A008668 A008669 A008670

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved