A005916 Molien series for a certain group of order 52.

1, 0, 1, 0, 2, 1, 3, 2, 5, 4, 7, 7, 11, 11, 15, 16, 21, 22, 28, 30, 37, 39, 47, 50, 60, 63, 74, 78, 91, 95, 109, 115, 131, 137, 154, 162, 181, 190, 210, 221, 243, 255, 278, 292, 318, 333, 360, 377, 407, 425, 457, 477, 512, 533, 570, 593, 633, 658, 700, 727

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 Molien series

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

Sequence in context: A238782 A058736 A097451 * A034392 A361392 A181531

Adjacent sequences: A005913 A005914 A005915 * A005917 A005918 A005919

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Eric M. Schmidt, Feb 02 2013

Status

approved