A309634 G.f.: x * Sum_{k>=1} x^k / (1 - x^k)^a(k).

0, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 5, 4, 1, 7, 1, 9, 8, 6, 1, 7, 6, 7, 8, 14, 1, 18, 1, 9, 12, 6, 23, 17, 1, 9, 17, 17, 1, 35, 1, 31, 41, 8, 1, 23, 29, 24, 12, 44, 1, 33, 47, 49, 30, 16, 1, 61, 1, 20, 120, 40, 84, 105, 1, 35, 23, 85, 1, 68, 1, 19, 115, 88, 151, 160, 1

Offset: 1

Author

Ilya Gutkovskiy, Aug 10 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1001 terms from Antti Karttunen)

Crossrefs

Cf. A028815 (positions of 1's), A157019, A309633.

Programs

Maple

a:= proc(n) option remember; uses numtheory;
      add(binomial((n-1)/d+a(d)-2, a(d)-1), d=divisors(n-1))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 27 2025

Mathematica

a[n_] := a[n] = SeriesCoefficient[x Sum[x^k/(1 - x^k)^a[k], {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 80}]
a[n_] := a[n] = Sum[Binomial[(n - 1)/d + a[d] - 2, a[d] - 1], {d, Divisors[n - 1]}]; a[1] = 0; Table[a[n], {n, 1, 80}]

PARI

seq(n)={my(v=vector(n)); v[2]=1; for(n=2, #v-1, v[n+1] = sumdiv(n, d, binomial(n/d + v[d] - 2, v[d] - 1))); v} \\ Andrew Howroyd, Aug 10 2019

Formula

a(1) = 0; a(n+1) = Sum_{d|n} binomial(n/d+a(d)-2,a(d)-1).