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

%I A309634 #16 Jan 27 2025 14:37:26
%S A309634 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,
%T A309634 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,
%U A309634 16,1,61,1,20,120,40,84,105,1,35,23,85,1,68,1,19,115,88,151,160,1
%N A309634 G.f.: x * Sum_{k>=1} x^k / (1 - x^k)^a(k).
%H A309634 Alois P. Heinz, <a href="/A309634/b309634.txt">Table of n, a(n) for n = 1..10000</a> (first 1001 terms from Antti Karttunen)
%F A309634 a(1) = 0; a(n+1) = Sum_{d|n} binomial(n/d+a(d)-2,a(d)-1).
%p A309634 a:= proc(n) option remember; uses numtheory;
%p A309634       add(binomial((n-1)/d+a(d)-2, a(d)-1), d=divisors(n-1))
%p A309634     end:
%p A309634 seq(a(n), n=1..80);  # _Alois P. Heinz_, Jan 27 2025
%t A309634 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}]
%t A309634 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}]
%o A309634 (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
%Y A309634 Cf. A028815 (positions of 1's), A157019, A309633.
%K A309634 nonn
%O A309634 1,5
%A A309634 _Ilya Gutkovskiy_, Aug 10 2019