A078411 Expansion of Molien series for a certain 4-D group of order 48.

1, 1, 3, 5, 10, 14, 23, 31, 46, 59, 80, 100, 130, 157, 196, 233, 283, 330, 392, 451, 527, 599, 689, 776, 883, 985, 1109, 1229, 1372, 1510, 1673, 1831, 2016, 2195, 2402, 2604, 2836, 3061, 3318, 3569, 3853, 4130, 4442, 4747, 5089, 5423, 5795, 6160, 6565, 6961, 7399

Offset

0,3

Comments

The first formula intersperses the terms with zeros, the second formula does not. - Colin Barker, Apr 02 2015

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for Molien series

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

Formula

G.f.: (1 +x^4 +x^6 +2*x^8 +x^10 +x^12 +x^16)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)), even powers only.

G.f.: (1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/ ((1-x)^4*(1+x)^2*(1+x^2)*(1+x+x^2)). - Colin Barker, Apr 02 2015

Example

G.f. = 1 + x^2 + 3*x^4 + 5*x^6 + 10*x^8 + 14*x^10 + 23*x^12 + 31*x^14 + 46*x^16 + ...

Maple

S:= series((1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/mul(1-x^j, j=1..4), x, 65):
seq(coeff(S, x, j), j = 0..60); # G. C. Greubel, Feb 02 2020

Mathematica

CoefficientList[Series[(1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)), {x, 0, 60}], x] (* G. C. Greubel, Feb 02 2020 *)

Prog

(Magma) // Definition of group: F<al> := CyclotomicField(24); M := GeneralLinearGroup(4, F);
B1 := M![ -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1 ]; C1 := M![1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0 ];
C2 := M![0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0 ]; G := sub<M | B1, C1, C2 >;
(PARI) Vec((x^8+x^6+x^5+2*x^4+x^3+x^2+1) / ((x-1)^4*(x+1)^2*(x^2+1)*(x^2+x+1)) + O(x^60)) \\ Colin Barker, Apr 02 2015
(Sage)
def A078411_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1 +x^2 +x^3 +2*x^4 +x^5 +x^6 +x^8)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) ).list()
A078411_list(60) # G. C. Greubel, Feb 02 2020

Crossrefs

Subgroup of the group in A078404.

Sequence in context: A365763 A008610 A281688 * A137630 A320886 A323433

Adjacent sequences: A078408 A078409 A078410 * A078412 A078413 A078414

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2002

Status

approved