A068491 Expansion of Molien series for a certain 4-D group of order 96.

1, 1, 2, 3, 6, 8, 13, 17, 25, 31, 42, 52, 68, 81, 101, 119, 145, 168, 200, 229, 268, 303, 349, 392, 447, 497, 560, 619, 692, 760, 843, 921, 1015, 1103, 1208, 1308, 1426, 1537, 1667, 1791, 1935, 2072, 2230, 2381, 2554, 2719, 2907, 3088, 3293, 3489, 3710, 3923, 4162

Offset

0,3

Comments

The first formula intersperses the terms with zeros, the second formula doesn't. - Colin Barker, Apr 01 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,-2,2,-2,1,0,-1,2,-1).

Formula

G.f.: (x^22+x^16+x^14+x^12+x^10+x^8+x^6+1)/((1-x^2)*(1-x^4)*(1-x^8)*(1-x^12)).

G.f.: (x^10-x^9+x^8+x^6+x^4+x^2-x+1) / ((x-1)^4*(x+1)^2*(x^2-x+1)*(x^2+1)*(x^2+x+1)). - Colin Barker, Apr 01 2015

Example

1 + x^2 + 2*x^4 + 3*x^6 + 6*x^8 + 8*x^10 + 13*x^12 + 17*x^14 + 25*x^16 + 31*x^18 + ...

Mathematica

LinearRecurrence[{2, -1, 0, 1, -2, 2, -2, 1, 0, -1, 2, -1}, {1, 1, 2, 3, 6, 8, 13, 17, 25, 31, 42, 52}, 60] (* Harvey P. Dale, Aug 29 2016 *)

Prog

(Magma) // Definition of group: F<al> := CyclotomicField(12); w := al^4; i := al^3; s3 := (1+2*w)/i; M := GeneralLinearGroup(4, F);
g1 := M![ 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0 ]; g2 := M![ -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0 ]; g3 := M![ 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1 ];
H := 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 ]; G := sub<M| g1, g2, g3, H>;
(PARI) Vec((x^10-x^9+x^8+x^6+x^4+x^2-x+1) / ((x-1)^4*(x+1)^2*(x^2-x+1)*(x^2+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Apr 01 2015

Crossrefs

Sequence in context: A319390 A251260 A022943 * A364796 A239952 A353902

Adjacent sequences: A068488 A068489 A068490 * A068492 A068493 A068494

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 31 2002

Status

approved