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A357041 a(n) = Sum_{d|n} 2^(d-1) * binomial(d+n/d-1,d).

%I A357041 #110 Jul 31 2023 02:25:36
%S A357041 1,4,7,18,21,66,71,196,305,648,1035,2526,4109,8774,16875,34288,65553,
%T A357041 134860,262163,531506,1051237,2109594,4194327,8425348,16779257,
%U A357041 33611984,67123631,134350206,268435485,537178750,1073741855,2148064768,4295048345,8591114580
%N A357041 a(n) = Sum_{d|n} 2^(d-1) * binomial(d+n/d-1,d).
%F A357041 G.f.: (1/2) * Sum_{k>0} (1/(1 - 2 * x^k)^k - 1).
%F A357041 G.f.: (1/2) * Sum_{k>0} (2 * x)^k/(1 - x^k)^(k+1).
%F A357041 If p is prime, a(p) = p + 2^(p-1).
%t A357041 a[n_] := DivisorSum[n, 2^(#-1) * Binomial[# + n/# - 1, #] &]; Array[a, 50] (* _Amiram Eldar_, Jul 31 2023 *)
%o A357041 (PARI) a(n) = sumdiv(n, d, 2^(d-1)*binomial(d+n/d-1, d));
%o A357041 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (1/(1-2*x^k)^k-1))/2)
%o A357041 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (2*x)^k/(1-x^k)^(k+1))/2)
%o A357041 (Python)
%o A357041 from math import comb
%o A357041 from sympy import divisors
%o A357041 def A357041(n): return sum(comb(d+n//d-1,d)<<d-1 for d in divisors(n,generator=True)) # _Chai Wah Wu_, Feb 27 2023
%Y A357041 Cf. A081543, A338682.
%Y A357041 Cf. A360797.
%K A357041 nonn
%O A357041 1,2
%A A357041 _Seiichi Manyama_, Feb 26 2023