A360802 Expansion of Sum_{k>0} (x / (1 - (2 * x)^k))^k.
1, 3, 5, 17, 17, 105, 65, 449, 641, 1953, 1025, 16257, 4097, 37761, 93185, 247809, 65537, 1499649, 262145, 6596609, 8847361, 13654017, 4194305, 210026497, 90177537, 251764737, 833880065, 2659418113, 268435457, 18345328641, 1073741825, 53553922049, 75438751745
Offset: 1
Keywords
Programs
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Mathematica
a[n_] := DivisorSum[n, 2^(n-#) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Aug 02 2023 *)
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PARI
my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (x/(1-(2*x)^k))^k)) -
PARI
a(n) = sumdiv(n, d, 2^(n-d)*binomial(d+n/d-2, d-1));
Formula
a(n) = Sum_{d|n} 2^(n-d) * binomial(d+n/d-2,d-1).
If p is prime, a(p) = 1 + 2^(p-1).