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A037240
Molien series for 3-D group X1.
6
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
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.
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
KEYWORD
nonn
EXTENSIONS
Terms a(35) and beyond from Alexander Karpov, Oct 29 2017
STATUS
approved