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A376016 a(n) = Sum_{d|n} d^(n/d) * binomial(n/d,d).

%I A376016 #19 Sep 06 2024 14:08:07
%S A376016 1,2,3,8,5,30,7,104,36,330,11,1296,13,2702,2445,7440,17,33030,19,
%T A376016 51220,76566,112662,23,699216,3150,639002,1653399,2064412,29,10620300,
%U A376016 31,12451872,29229288,17825826,1640660,190101888,37,89653286,455976417,441305440,41
%N A376016 a(n) = Sum_{d|n} d^(n/d) * binomial(n/d,d).
%H A376016 Seiichi Manyama, <a href="/A376016/b376016.txt">Table of n, a(n) for n = 1..5000</a>
%F A376016 G.f.: Sum_{k>=1} (k*x^k)^k / (1 - k*x^k)^(k+1).
%F A376016 If p is prime, a(p) = p.
%o A376016 (PARI) a(n) = sumdiv(n, d, d^(n/d)*binomial(n/d, d));
%o A376016 (PARI) my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, (k*x^k)^k/(1-k*x^k)^(k+1)))
%o A376016 (Python)
%o A376016 from math import comb
%o A376016 from itertools import takewhile
%o A376016 from sympy import divisors
%o A376016 def A376016(n): return sum(d**(m:=n//d)*comb(m,d) for d in takewhile(lambda d:d**2<=n,divisors(n))) # _Chai Wah Wu_, Sep 06 2024
%Y A376016 Cf. A055225, A327238, A376020.
%K A376016 nonn
%O A376016 1,2
%A A376016 _Seiichi Manyama_, Sep 06 2024