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A008743 Molien series for 3-dimensional group [3,4]+ = 432. 0
1, 0, 1, 0, 2, 0, 3, 0, 4, 1, 5, 1, 7, 2, 8, 3, 10, 4, 12, 5, 14, 7, 16, 8, 19, 10, 21, 12, 24, 14, 27, 16, 30, 19, 33, 21, 37, 24, 40, 27, 44, 30, 48, 33, 52, 37, 56, 40, 61, 44, 65, 48, 70, 52, 75, 56, 80, 61, 85, 65, 91, 70, 96, 75, 102, 80, 108, 85, 114 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The binary quintic has four invariants of degrees 4, 8, 12, 18. Those of degrees 4, 8, 12 are algebraically independent, the one of degree 18 squares to an expression in the others. [A. E. Brouwer]

LINKS

Table of n, a(n) for n=0..68.

A. E. Brouwer, Invariants of binary forms

Marko V. Jaric and Joseph L. Birman, Calculation of the Molien generating function for invariants of space groups, J. Math. Phys. 18 (1977), 1459-1465.

Index entries for Molien series

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

FORMULA

Euler transform of length 18 sequence [ 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Oct 30 2011

G.f.: (1 + x^9) / ((1 - x^2) * (1 - x^4) * (1 - x^6)). a(-3 - n) = a(n).

a(2*n) = A001399(n). a(2*n + 1) = A001399(n - 4). - Michael Somos, Oct 30 2011

G.f.: ( -1+x^3-x^6 ) / ( (1+x+x^2)*(1+x^2)*(1+x)^2*(x-1)^3 ). - R. J. Mathar, Dec 18 2014

EXAMPLE

1 + x^2 + 2*x^4 + 3*x^6 + 4*x^8 + x^9 + 5*x^10 + x^11 + 7*x^12 + 2*x^13 + 8*x^14 + ...

1 + q^4 + 2*q^8 + 3*q^12 + 4*q^16 + q^18 + 5*q^20 + q^22 + 7*q^24 + 2*q^26 + 8*q^28 + ...

MAPLE

(1+x^9)/(1-x^2)/(1-x^4)/(1-x^6);

MATHEMATICA

CoefficientList[Series[(1 + x^9)/((1 - x^2)*(1 - x^4)*(1 - x^6)), {x, 0, 100}], x] (* T. D. Noe, Oct 30 2011 *)

PROG

(PARI) {a(n) = round( (if( n%2, n-9, n) \ 2 + 3)^2 / 12)} /* Michael Somos, Oct 30 2011 */

(PARI) {a(n) = if( n<-1, n = -3 - n);  polcoeff( (1+x^9)/(1-x^2)/(1-x^4)/(1-x^6) + x * O(x^n), n)} /* Michael Somos, Oct 30 2011 */

CROSSREFS

Cf. A001399.

Sequence in context: A115720 A053120 A284976 * A029179 A008721 A008735

Adjacent sequences:  A008740 A008741 A008742 * A008744 A008745 A008746

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified January 17 19:58 EST 2019. Contains 319251 sequences. (Running on oeis4.)