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A221578 A sum over partitions (q=6), see first comment.

%I A221578 #19 Feb 28 2022 01:53:18
%S A221578 1,5,35,210,1290,7735,46620,279685,1679370,10076190,60464670,
%T A221578 362787810,2176773305,13060638360,78364108620,470184650495,
%U A221578 2821109573550,16926657432510,101559954663930,609359727929610,3656158427989830,21936950567886270,131621703769781995
%N A221578 A sum over partitions (q=6), see first comment.
%C A221578 Set q=6 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 A221578 Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
%C A221578 q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
%C A221578 q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
%C A221578 q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
%C A221578 Sequences where q is not a prime power:
%C A221578 q=6: A221578, q=10: A221579, q=12: A221580,
%C A221578 q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.
%H A221578 Alois P. Heinz, <a href="/A221578/b221578.txt">Table of n, a(n) for n = 0..400</a>
%p A221578 with(numtheory):
%p A221578 b:= proc(n) b(n):= add(phi(d)*6^(n/d), d=divisors(n))/n-1 end:
%p A221578 a:= proc(n) a(n):= `if`(n=0, 1,
%p A221578        add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
%p A221578     end:
%p A221578 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 24 2013
%t A221578 b[n_] := Sum[EulerPhi[d]*6^(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 A221578 (PARI)
%o A221578 N=66; x='x+O('x^N);
%o A221578 gf=prod(n=1,N, (1-x^n)/(1-6*x^n)  );
%o A221578 v=Vec(gf)
%K A221578 nonn
%O A221578 0,2
%A A221578 _Joerg Arndt_, Jan 20 2013