A037240 Molien series for 3-D group X1.

1, 1, 5, 10, 24, 42, 83, 132, 222, 335, 511, 728, 1047, 1428, 1956, 2586, 3414, 4389, 5638, 7084, 8888, 10966, 13494, 16380, 19841, 23751, 28371, 33566, 39616, 46376, 54177, 62832, 72726, 83661, 96045, 109668, 124999, 141778, 160538, 181006, 203742, 228459, 255788, 285384

Offset

0,3

Comments

Also multidigraphs with 3 nodes and n arcs. - Vladeta Jovovic, Dec 27 1999

Also preference profiles with 3 alternatives and n agents (IANC model). - Alexander Karpov, Nov 23 2017

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Ö. Egecioglu, Uniform generation of anonymous and neutral preference profiles for social choice rules, Technical Report, UCSB, 2005.

Ö. Egecioglu, Uniform generation of anonymous and neutral preference profiles for social choice rules, Monte Carlo Methods and Applications, 15(3), Jan 2009, 241-255.

Ira Gessel, Combinatorial counting with symmetry, MathOverflow, 2014.

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; 2085.

Alexander V. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.

Index entries for Molien series

Formula

G.f.: (1 + x^2 + 3*x^3 + 5*x^4 + x^5 + x^6)/((1 - x)*(1 - x^2)^3*(1 - x^3)^2).

From Alexander Karpov, Nov 18 2017: (Start)

if n == 0 mod 6, a(n) = C(n+5,5)/6 + (n+4)*(n+2)/16 + (n+3)/9;

if n == 3 mod 6, a(n) = C(n+5,5)/6 + (n+3)/9;

if n == 2,4 mod 6, a(n) = C(n+5,5)/6 + (n+4)*(n+2)/16;

if n == 1,5 mod 6, a(n) = C(n+5,5)/6.

(End)

Maple

S:= series((1+x^2+3*x^3+5*x^4+x^5+ x^6)/(1 - x)/(1 - x^2)^3/(1 - x^3)^2, x, 101):
seq(coeff(S, x, n), n=0..100); # Robert Israel, Nov 22 2017

Mathematica

CoefficientList[Series[(1 +x^2 +3x^3 +5x^4 +x^5 +x^6)/(1-x)/(1-x^2)^3/(1-x^3)^2, {x, 0, 43}], x] (* Michael De Vlieger, Nov 01 2017 *)

Prog

(PARI) Vec((1+x^2+3*x^3+5*x^4+x^5+x^6)/(1-x)/(1-x^2)^3/(1-x^3)^2 + O(x^50)) \\ Michel Marcus, Oct 31 2017
(Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1 +x^2 +3*x^3 +5*x^4 +x^5 +x^6)/((1-x)*(1-x^2)^3*(1-x^3)^2) )); // G. C. Greubel, Jan 31 2020
(Sage)
def A037240_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2+3*x^3+5*x^4+x^5+x^6)/((1-x)*(1-x^2)^3*(1-x^3)^2) ).list()
A037240_list(50) # G. C. Greubel, Jan 31 2020

Crossrefs

Column k=3 of A333361.

Sequence in context: A280721 A300552 A358259 * A182095 A177432 A269740

Adjacent sequences: A037237 A037238 A037239 * A037241 A037242 A037243

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Terms a(35) and beyond from Alexander Karpov, Oct 29 2017

Status

approved