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

Author

N. J. A. Sloane

Keywords

• nonn

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

Crossrefs

Column k=3 of A333361.

Programs

Magma

R:=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

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 *)

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

Sage

def A037240_list(prec):
    P. = 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

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)

Extensions

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