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Expansion of Sum_{k>0} x^k / (1 - k * x^k)^(k+1).
3

%I #15 Jul 31 2023 02:25:05

%S 1,3,4,11,6,43,8,109,100,281,12,1507,14,1863,3376,6937,18,26245,20,

%T 53211,63022,67739,24,572413,78776,372945,1087048,1761719,30,7362871,

%U 32,9947953,16897486,10027349,8011116,123101515,38,49807779,241823440,361722421,42

%N Expansion of Sum_{k>0} x^k / (1 - k * x^k)^(k+1).

%F a(n) = Sum_{d|n} d^(n/d-1) * binomial(d+n/d-1,d).

%F If p is prime, a(p) = 1 + p.

%t a[n_] := DivisorSum[n, #^(n/# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* _Amiram Eldar_, Jul 31 2023 *)

%o (PARI) my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-k*x^k)^(k+1)))

%o (PARI) a(n) = sumdiv(n, d, d^(n/d-1)*binomial(d+n/d-1, d));

%Y Cf. A360782, A360783.

%Y Cf. A081543, A324158.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Feb 21 2023