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A360733 Expansion of Sum_{k>0} (x * (1 + (k * x)^k))^k.

%I A360733 #19 Aug 09 2023 00:51:47
%S A360733 1,2,1,9,1,98,1,1025,2188,15626,1,692836,1,5764802,97656251,201326593,
%T A360733 1,36138519442,1,409470748547,14242684529830,3138428376722,1,
%U A360733 10019491686645761,476837158203126,3937376385699290,5403406870691968357,19704673338472752470,1
%N A360733 Expansion of Sum_{k>0} (x * (1 + (k * x)^k))^k.
%H A360733 Winston de Greef, <a href="/A360733/b360733.txt">Table of n, a(n) for n = 1..599</a>
%F A360733 a(n) = Sum_{d|n} d^(n-d) * binomial(d,n/d-1).
%F A360733 If p is an odd prime, a(p) = 1.
%t A360733 a[n_] := DivisorSum[n, #^(n-#) * Binomial[#, n/# - 1] &]; Array[a, 30] (* _Amiram Eldar_, Aug 09 2023 *)
%o A360733 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, (x*(1+(k*x)^k))^k))
%o A360733 (PARI) a(n) = sumdiv(n, d, d^(n-d)*binomial(d, n/d-1));
%Y A360733 Cf. A327249, A360732.
%K A360733 nonn
%O A360733 1,2
%A A360733 _Seiichi Manyama_, Feb 18 2023