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 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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved