A360797 -id:A360797 - cp's OEIS frontend

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

1, 4, 7, 18, 21, 66, 71, 196, 305, 648, 1035, 2526, 4109, 8774, 16875, 34288, 65553, 134860, 262163, 531506, 1051237, 2109594, 4194327, 8425348, 16779257, 33611984, 67123631, 134350206, 268435485, 537178750, 1073741855, 2148064768, 4295048345, 8591114580

Offset: 1

Seiichi Manyama, Feb 26 2023

nonn

Cf. A081543, A338682.

Cf. A360797.

a[n_] := DivisorSum[n, 2^(#-1) * Binomial[# + n/# - 1, #] &]; Array[a, 50] (* Amiram Eldar, Jul 31 2023 *)

a(n) = sumdiv(n, d, 2^(d-1)*binomial(d+n/d-1, d));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (1/(1-2*x^k)^k-1))/2)

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (2*x)^k/(1-x^k)^(k+1))/2)

from math import comb
from sympy import divisors
def A357041(n): return sum(comb(d+n//d-1,d)<Chai Wah Wu, Feb 27 2023

G.f.: (1/2) * Sum_{k>0} (1/(1 - 2 * x^k)^k - 1).
G.f.: (1/2) * Sum_{k>0} (2 * x)^k/(1 - x^k)^(k+1).
If p is prime, a(p) = p + 2^(p-1).

A360798 Expansion of Sum_{k>0} x^k / (1 - (2 * x)^k)^(k+1).

Original entry on oeis.org

1, 5, 13, 45, 81, 321, 449, 1745, 2945, 9153, 11265, 60609, 53249, 230401, 410625, 1259777, 1114113, 7263233, 4980737, 31337473, 44630017, 115367937, 96468993, 937283585, 551550977, 2399256577, 4594597889, 14579646465, 7784628225, 89894944769, 33285996545

Offset: 1

Seiichi Manyama, Feb 21 2023

nonn

Cf. A360797.

a[n_] := DivisorSum[n, 2^(n-#) * Binomial[# + n/# - 1, #] &]; Array[a, 30] (* Amiram Eldar, Aug 02 2023 *)

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(2*x)^k)^(k+1)))

a(n) = sumdiv(n, d, 2^(n-d)*binomial(d+n/d-1, d));

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

If p is prime, a(p) = 1 + p * 2^(p-1).

cp's OEIS Frontend

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

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A360798 Expansion of Sum_{k>0} x^k / (1 - (2 * x)^k)^(k+1).

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Mathematica

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