OFFSET
0,5
COMMENTS
The group is a semidirect product C13: C4 presented by <g, h | g^13=1, h^4=1, hg = g^5 h>. The group has 3 irreducible characters of degree 4, all of which have the same Molien series, this sequence. - Eric M. Schmidt, Feb 02 2013
LINKS
Eric M. Schmidt, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1,0,0,0,0,0,1,-1,-1,1,-1,1,1,-1).
FORMULA
G.f.: (1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)). - Colin Barker, Jan 31 2013, confirmed and simplified by Eric M. Schmidt, Feb 02 2013
a(n) ~ (1/312)*n^3. - Ralf Stephan, Apr 29 2014
MAPLE
m:=60; S:=series((1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 06 2020
MATHEMATICA
CoefficientList[Series[(1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)), {x, 0, 60}], x] (* G. C. Greubel, Feb 06 2020 *)
LinearRecurrence[{1, 1, -1, 1, -1, -1, 1, 0, 0, 0, 0, 0, 1, -1, -1, 1, -1, 1, 1, -1}, {1, 0, 1, 0, 2, 1, 3, 2, 5, 4, 7, 7, 11, 11, 15, 16, 21, 22, 28, 30}, 60] (* Harvey P. Dale, May 11 2022 *)
PROG
(GAP) series:=MolienSeries(First(Irr(SmallGroup(52, 3)), irr->Degree(irr)=4));; List([0..30], i->ValueMolienSeries(series, i)); # Eric M. Schmidt, Feb 02 2013
(PARI) Vec( (1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)) +O('x^60) ) \\ G. C. Greubel, Feb 06 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)) )); // G. C. Greubel, Feb 06 2020
(Sage)
def A005916_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-x+x^5+x^11-x^13+x^14)/((1-x)*(1-x^2)*(1-x^4)*(1-x^13)) ).list()
A005916_list(60) # G. C. Greubel, Feb 06 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Eric M. Schmidt, Feb 02 2013
STATUS
approved