A008614 Molien series of 3-dimensional representation of group GL(3,2) (= L(2,7)); a simple group of order 168.

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 3, 0, 3, 1, 3, 0, 4, 1, 4, 1, 5, 1, 5, 1, 6, 2, 6, 2, 7, 2, 7, 3, 8, 3, 9, 3, 9, 4, 10, 4, 11, 5, 11, 5, 12, 6, 13, 6, 14, 7, 14, 7, 16, 8, 16, 9, 17, 9, 18, 10, 19, 11, 20, 11, 21, 12, 22, 13, 23, 14, 24, 14, 25, 16, 26, 16, 28, 17, 28, 18, 30

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

The simple group of order 168 expressed as a group of linear substitutions on three variables has invariants of degrees 4, 6, 14 which are rationally independent. The invariant of degree 4 is x1*x2^3 + x2*x3^3 + x3*x1^3 (Klein's quartic curve). - Michael Somos, Mar 18 2015

Examples

			G.f. = 1 + x^4 + x^6 + x^8 + x^10 + 2*x^12 + 2*x^14 + 2*x^16 + 3*x^18 + ...
		

References

• D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 101.
• W. Burnside, Theory of Groups of Finite Order, Dover Publications, NY, 1955, section 267, page 363. There is a typo in his formula: the term with numerator 21 should have denominator (1+x)(1-x^2). [Added by N. J. A. Sloane, Mar 01 2012]
• T. A. Springer, Invariant Theory, Lecture Notes in Math., Vol. 585, Springer, p. 97.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (-1,0,1,2,1,0,0,0,0,-1,-2,-1,0,1,1).

Crossrefs

Cf. A008671.

Programs

Maple

(1+x^21)/(1-x^4)/(1-x^6)/(1-x^14);

Mathematica

LinearRecurrence[{-1,0,1,2,1,0,0,0,0,-1,-2,-1,0,1,1},{1,0,0,0,1,0,1,0,1,0,1,0,2,0,2},100] (* Harvey P. Dale, Jan 17 2015 *)
a[ n_] := Module[{m = If[ n < 0, -3 - n, n]}, m = If[ OddQ[m], m - 21, m] / 2; SeriesCoefficient[ 1 / ((1 - x^2) (1 - x^3) (1 - x^7)), {x, 0, m}]]; (* Michael Somos, Mar 18 2015 *)

PARI

{a(n) = if( n%2, n-=21); n/=2; if( n<-11, n=-12-n); polcoeff( 1 / ((1-x^2) * (1-x^3) * (1-x^7)) + x * O(x^n), n)}; /* Michael Somos, Oct 11 2006 */

Formula

Euler transform of length 42 sequence [0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Oct 11 2006

G.f.: (1 - x^42) / ((1 - x^4) * (1 - x^6) * (1 - x^14) * (1 - x^21)). - Michael Somos, Oct 11 2006

a(n) = a(-3 - n). a(n) = a(n-4) + a(n-6) - a(n-10) + a(n-14) - a(n-18) - a(n-20) + a(n-24) for all n in Z. - Michael Somos, Oct 11 2006

a(2*n + 21) = a(2*n) = A008671(n) for all n in Z.

a(n) ~ 1/336*n^2. - Ralf Stephan, Apr 29 2014

a(n)= -a(n-1) +a(n-3) +2*a(n-4) +a(n-5) -a(n-10) -2*a(n-11) -a(n-12) +a(n-14) +a(n-15). - R. J. Mathar, Dec 18 2014

G.f.: (1/168) * ( 1 / (1 - x)^3 + 21 / ((1 + x) * (1 - x^2)) + 56 / (1 - x^3) + 42 / ((1 - x) * (1 + x^2)) + 24 * (1 - x) * (2 + 3*x + 2*x^2) / (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). [Burnside] - Michael Somos, Mar 18 2015