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Expansion of Sum_{k>0} x^k / (1 - (k * x)^k)^(k+1).
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%I #15 Aug 02 2023 02:00:05

%S 1,3,4,17,6,211,8,1929,7300,22601,12,1724809,14,6703047,223678576,

%T 738787345,18,65630598229,20,2119646503661,24448573943662,

%U 3423809253371,24,21453113652593665,12016296386718776,4240253019018225,8255251542208471048,67251293544533119589,30

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

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

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

%t a[n_] := DivisorSum[n, #^(n-#) * Binomial[# + n/# - 1, #] &]; Array[a, 30] (* _Amiram Eldar_, Aug 02 2023 *)

%o (PARI) my(N=30, 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)*binomial(d+n/d-1, d));

%Y Cf. A360787, A360788.

%Y Cf. A339481, A360794.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Feb 21 2023