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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 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 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 STATUS approved

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