A008668 Molien series for 4-dimensional reflection group [3,3,5] of order 14400.

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 32, 33, 36, 37, 38, 41, 44, 45, 48, 49, 52, 55, 58, 59, 62, 65, 68, 71, 74, 75, 81, 84, 87, 90, 93, 96, 102, 105, 108, 111, 117, 120, 126, 129, 132, 138

Offset

0,7

Comments

The relevant generating function is 1/((1-z^2)*(1-z^12)*(1-z^20)*(1-z^30)) and is reduced with x=z^2 below to indicate that the intermediate zeros are not listed.

Number of partitions into parts 1, 6, 10, and 15. - Joerg Arndt, Apr 29 2014

References

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, Ergebnisse der Mathematik und Ihrer Grenzgebiete, New Series, no. 14. Springer Verlag, 1957, Table 10.

L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 30).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Roberto De Maria Nunes Mendes, Symmetries of spherical harmonics, Transactions of the American Mathematical Society 204 (1975): 161-178. See subgroup 68.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 240

Index entries for Molien series

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1).

Formula

G.f.: 1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15)). - M. F. Hasler, Mar 26 2012

a(n) ~ 1/5400*n^3. - Ralf Stephan, Apr 29 2014

Maple

seq(coeff(series(1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15)), x, n+1), x, n), n = 0 .. 80); # G. C. Greubel, Sep 08 2019

Mathematica

CoefficientList[Series[1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15)), {x, 0, 80}], x] (* G. C. Greubel, Sep 08 2019 *)

Prog

(PARI) A008668_list = n -> Vec(1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15)) +O(x^n)) \\ M. F. Hasler, Mar 26 2012
(Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15)) )); // G. C. Greubel, Sep 08 2019
(Sage)
def A008668_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)*(1-x^6)*(1-x^10)*(1-x^15))).list()
A008668_list(80) # G. C. Greubel, Sep 08 2019

Crossrefs

Sequence in context: A056970 A212218 A321162 * A225643 A116563 A076695

Adjacent sequences: A008665 A008666 A008667 * A008669 A008670 A008671

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Terms a(61) onward added by G. C. Greubel, Sep 08 2019

Status

approved