A049315 - OEIS

%I #33 Sep 08 2022 08:44:58

%S 1,4,24,120,620,3096,15600,77976,390480,1952380,9764880,48824280,

%T 244136904,1220683800,6103496400,30517481424,152587794020,

%U 762938966520,3814696782120,19073483892120,95367429207720,476837146020720,2384185778835696,11920928894086200

%N The number k(GL(n,q)) of conjugacy classes in GL(n,q), q=5.

%D V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

%H Alois P. Heinz, <a href="/A049315/b049315.txt">Table of n, a(n) for n = 0..450</a>

%H W. Feit and N. J. Fine, <a href="https://projecteuclid.org/euclid.dmj/1077468920">Pairs of commuting matrices over a finite field</a>, Duke Math. Journal, 27 (1960) 91-94.

%F The number a(n) of conjugacy classes in the group GL(n, q) is the coefficient of t^n in the infinite product: product k=1, 2, ... (1-t^k)/(1-qt^k) - Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001.

%F G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(5^(k/d) - 1) ) * x^k/k). - _Ilya Gutkovskiy_, Sep 27 2018

%p with(numtheory):

%p b:= proc(n) b(n):= add(phi(d)*5^(n/d), d=divisors(n))/n-1 end:

%p a:= proc(n) a(n):= `if`(n=0, 1,

%p        add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 03 2012

%t b[n_] := Sum[EulerPhi[d]*5^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jan 24 2014, after _Alois P. Heinz_ *)

%o (Magma) /* The program does not work for n>8: */ [1] cat [NumberOfClasses(GL(n,5)) : n in [1..8]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by _Vincenzo Librandi_, Jan 23 2013

%o (PARI) x='x+O('x^30); Vec(prod(n=1, 30, (1-x^n)/(1-5*x^n))) \\ _Altug Alkan_, Sep 27 2018

%Y Cf. A006951, A006952, A049314, A049316.

%K nonn

%O 0,2

%A _Vladeta Jovovic_