A078404 Expansion of Molien series for a certain 4-D group of order 192.

1, 1, 1, 2, 4, 5, 7, 9, 14, 17, 22, 27, 36, 43, 52, 61, 75, 87, 102, 116, 137, 155, 177, 198, 227, 253, 283, 312, 350, 385, 425, 463, 512, 557, 608, 657, 718, 775, 838, 899, 973, 1043, 1120, 1194, 1283, 1367, 1459, 1548, 1653, 1753, 1861, 1966, 2088, 2205, 2331, 2453

Offset

0,4

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 (2,-1,0,1,-2,1,0,0,0,0,0,1,-2,1,0,-1,2,-1).

Formula

G.f.: (1 +x^8 +x^10 +x^12 +2*x^16 +2*x^20 +x^24 +x^26 +x^28 +x^36)/((1-x^2)*(1-x^6)*(1-x^8)*(1-x^24)), even terms only.

G.f.: (1 -x +x^3 +x^6 -x^7 +2*x^8 -x^9 +x^10 +x^13 -x^15 +x^16)/ ((1-x)^4* (1+x)^2*(1-x+x^2)*(1+x^2)^2*(1+x+x^2)*(1-x^2+x^4)). - Colin Barker, Apr 02 2015

Example

G.f. = 1 + x^2 + x^4 + 2*x^6 + 4*x^8 + 5*x^10 + 7*x^12 + ...

Maple

S:=series( (1 +x^4 +x^5 +x^6 +2*x^8 +2*x^10 +x^12 +x^13 +x^14 +x^18 )/((1-x)*(1-x^3)*(1-x^4)*(1-x^12)), x, 65): seq(coeff(S, x, j), j=0..60); # G. C. Greubel, Feb 02 2020

Mathematica

CoefficientList[Series[(1 +x^4 +x^5 +x^6 +2*x^8 +2*x^10 +x^12 +x^13 +x^14 +x^18 )/((1-x)*(1-x^3)*(1-x^4)*(1-x^12)), {x, 0, 60}], x] (* G. C. Greubel, Feb 02 2020 *)

Prog

(Magma) // Definition of group:
F<al> := CyclotomicField(24); i := al^6; eta := al^3; s2 := (1+i)/eta; om := al^8; s3 := (2*om+1)/i; s6 := s2*s3; M := GeneralLinearGroup(4, F);
A1 := M![s3/s6, 1/s6, 1/s6, 1/s6, -1/s6, s3/s6, -1/s6, 1/s6, -1/s6, 1/s6, s3/s6, -1/s6, -1/s6, -1/s6, 1/s6, s3/s6 ]; A2 := M![0, 1/s3, 1/s3, 1/s3, -1/s3, 0, -1/s3, 1/s3, -1/s3, 1/s3, 0, -1/s3, -1/s3, -1/s3, 1/s3, 0 ];
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 | A1, A2, B1, C1, C2 >;
(PARI) Vec((x^16-x^15+x^13+x^10-x^9+2*x^8-x^7+x^6+x^3-x+1) / ((x-1)^4*(x+1)^2*(x^2-x+1)*(x^2+1)^2*(x^2+x+1)*(x^4-x^2+1)) + O(x^60)) \\ Colin Barker, Apr 02 2015
(Sage)
def A078404_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1 +x^4 +x^5 +x^6 +2*x^8 +2*x^10 +x^12 +x^13 +x^14 +x^18 )/((1-x)*(1-x^3)*(1-x^4)*(1-x^12)) ).list()
A078404_list(60) # G. C. Greubel, Feb 02 2020

Crossrefs

The group in A078411 is a subgroup.

Sequence in context: A193494 A161832 A350592 * A056908 A296344 A060565

Adjacent sequences: A078401 A078402 A078403 * A078405 A078406 A078407

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 27 2002

Status

approved