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%I #40 Aug 11 2024 05:51:48
%S 1,3,15,60,252,1005,4080,16305,65460,261828,1048260,4192980,16775955,
%T 67103520,268430160,1073720415,4294945932,17179782540,68719391100,
%U 274877559420,1099511281260,4398045120300,17592184654365,70368738597600,281474971147680
%N The number k(GL(n,q)) of conjugacy classes in GL(n,q), q=4.
%C Bound: k(GL(n,q))<q^n. Asymptotics: k(GL(n,q))~q^n as n tends to infinity.
%D Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
%H Alois P. Heinz, <a href="/A049314/b049314.txt">Table of n, a(n) for n = 0..500</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*(4^(k/d) - 1) ) * x^k/k). - _Ilya Gutkovskiy_, Sep 27 2018
%p with(numtheory):
%p b:= proc(n) b(n):= add(phi(d)*4^(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]*4^(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>9: */ [1] cat [NumberOfClasses(GL(n,4)) : 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)/(1-4*x^n))) \\ _Altug Alkan_, Sep 27 2018
%Y Cf. A006951, A006952, A049315, A049316.
%K nonn
%O 0,2
%A _Vladeta Jovovic_
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