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%I #14 Aug 02 2023 02:00:08
%S 1,5,13,45,81,321,449,1745,2945,9153,11265,60609,53249,230401,410625,
%T 1259777,1114113,7263233,4980737,31337473,44630017,115367937,96468993,
%U 937283585,551550977,2399256577,4594597889,14579646465,7784628225,89894944769,33285996545
%N Expansion of Sum_{k>0} x^k / (1 - (2 * x)^k)^(k+1).
%F a(n) = Sum_{d|n} 2^(n-d) * binomial(d+n/d-1,d).
%F If p is prime, a(p) = 1 + p * 2^(p-1).
%t a[n_] := DivisorSum[n, 2^(n-#) * Binomial[# + n/# - 1, #] &]; Array[a, 30] (* _Amiram Eldar_, Aug 02 2023 *)
%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(2*x)^k)^(k+1)))
%o (PARI) a(n) = sumdiv(n, d, 2^(n-d)*binomial(d+n/d-1, d));
%Y Cf. A360797.
%K nonn
%O 1,2
%A _Seiichi Manyama_, Feb 21 2023
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