A360755 Expansion of (1/2) * Sum_{k>0} (2 * x * (1 + x^k))^k.
1, 3, 4, 12, 16, 46, 64, 160, 268, 592, 1024, 2292, 4096, 8640, 16544, 33824, 65536, 133856, 262144, 529576, 1049920, 2108416, 4194304, 8417408, 16777296, 33607680, 67118080, 134334656, 268435456, 537140208, 1073741824, 2148015104, 4295023616, 8591048704
Offset: 1
Keywords
Programs
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Mathematica
a[n_] := DivisorSum[n, 2^(#-1) * Binomial[#, n/# - 1] &]; Array[a, 35] (* Amiram Eldar, Aug 02 2023 *)
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PARI
my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (2*x*(1+x^k))^k)/2) -
PARI
a(n) = sumdiv(n, d, 2^(d-1)*binomial(d, n/d-1));
Formula
a(n) = Sum_{d|n} 2^(d-1) * binomial(d,n/d-1).
If p is an odd prime, a(p) = 2^(p-1).