login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A008667
Expansion of g.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).
14
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).
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
KEYWORD
nonn,easy
STATUS
approved