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%I #25 Jun 19 2023 12:12:13
%S 1,18,360,6840,130302,2475720,47045520,893864520,16983555840,
%T 322687560618,6131066120640,116490256285320,2213314916460120,
%U 42052983412605480,799006685733239040,15181127028931412160,288441413566677788022,5480386857766875373560
%N Number of conjugacy classes in GL(n,19).
%H Alois P. Heinz, <a href="/A182609/b182609.txt">Table of n, a(n) for n = 0..250</a>
%F G.f.: Product_{k>=1} (1-x^k)/(1-19*x^k). - _Alois P. Heinz_, Nov 03 2012
%p with(numtheory):
%p b:= proc(n) b(n):= add(phi(d)*19^(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]*19^(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_, Feb 17 2014, after _Alois P. Heinz_ *)
%o (Magma) /* The program does not work for n>4: */ [1] cat [NumberOfClasses(GL(n, 19)) : n in [1..4]];
%o (PARI)
%o N=66; x='x+O('x^N);
%o gf=prod(n=1,N, (1-x^n)/(1-19*x^n) );
%o v=Vec(gf)
%o /* _Joerg Arndt_, Jan 24 2013 */
%Y Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182604, A182605, A182606, A182607, A182608, A182610, A182611, A182612.
%K nonn
%O 0,2
%A _Klaus Brockhaus_, Nov 23 2010
%E More terms from _Alois P. Heinz_, Nov 03 2012
%E MAGMA code edited by _Vincenzo Librandi_, Jan 24 2013
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