A221579 A sum over partitions (q=10), see first comment.

1, 9, 99, 990, 9990, 99891, 999900, 9998901, 99998910, 999989010, 9999989010, 99999889110, 999999890109, 9999998890200, 99999998891100, 999999988901199, 9999999988902090, 99999999888912090, 999999999889011990, 9999999998889021990

Offset: 0

Author

Joerg Arndt, Jan 20 2013

Keywords

• nonn

Comments

Set q=10 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].

Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":

q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,

q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,

q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.

Sequences where q is not a prime power:

q=6: A221578, q=10: A221579, q=12: A221580,

q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Programs

Maple

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*10^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 25 2013

Mathematica

b[n_] := Sum[EulerPhi[d]*10^(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 *)

PARI

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-10*x^n)  );
v=Vec(gf)