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A221584 A sum over partitions (q=20), see first comment.

%I A221584 #12 Feb 24 2022 08:45:22
%S A221584 1,19,399,7980,159980,3199581,63999600,1279991601,25599991620,
%T A221584 511999832020,10239999832020,204799996632420,4095999996640419,
%U A221584 81919999932640800,1638399999932648400,32767999998652808799,655359999998652816380,13107199999973052976380
%N A221584 A sum over partitions (q=20), see first comment.
%C A221584 Set q=20 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P=[p_1^m_1, p_2^m_2, ..., p_L^m_L].
%C A221584 Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
%C A221584   q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
%C A221584   q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
%C A221584   q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
%C A221584 Sequences where q is not a prime power:
%C A221584   q=6: A221578, q=10: A221579, q=12: A221580,
%C A221584   q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.
%H A221584 Vincenzo Librandi, <a href="/A221584/b221584.txt">Table of n, a(n) for n = 0..100</a>
%o A221584 (PARI)
%o A221584 N=66; x='x+O('x^N);
%o A221584 gf=prod(n=1,N, (1-x^n)/(1-20*x^n)  );
%o A221584 v=Vec(gf)
%K A221584 nonn
%O A221584 0,2
%A A221584 _Joerg Arndt_, Jan 20 2013