A338693
a(n) = Sum_{d|n} d^(d - n/d) * binomial(d, n/d).
Original entry on oeis.org
1, 4, 27, 257, 3125, 46665, 823543, 16777312, 387420490, 10000001250, 285311670611, 8916100467712, 302875106592253, 11112006825910963, 437893890380859625, 18446744073716891649, 827240261886336764177, 39346408075296709766628, 1978419655660313589123979
Offset: 1
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a[n_] := DivisorSum[n, #^(# - n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
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a(n) = sumdiv(n, d, d^(d-n/d)* binomial(d, n/d));
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N=20; x='x+O('x^N); Vec(sum(k=1, N, (k+x^k)^k-k^k))
A338684
a(n) = Sum_{d|n} (-1)^(d-1) * (n/d)^n * binomial(d+n/d-1, d).
Original entry on oeis.org
1, 7, 82, 975, 15626, 275817, 5764802, 133561087, 3486981232, 99853521768, 3138428376722, 106947820494048, 3937376385699290, 155549105311903523, 6568409424129452048, 295137771929866797055, 14063084452067724991010, 708228596784096039676230, 37589973457545958193355602
Offset: 1
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a[n_] := DivisorSum[n, (-1)^(# - 1) * (n/#)^n * Binomial[# + n/# - 1, #] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
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a(n) = sumdiv(n, d, (-1)^(d-1)*(n/d)^n*binomial(d+n/d-1, d));
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N=20; x='x+O('x^N); Vec(sum(k=1, N, 1-1/(1+(k*x)^k)^k))
A376015
a(n) = Sum_{d|n} d^n * binomial(n/d,d).
Original entry on oeis.org
1, 2, 3, 20, 5, 198, 7, 1544, 19692, 10250, 11, 2187216, 13, 344078, 143489085, 4296802320, 17, 7757846982, 19, 5497605324820, 366112362126, 230686742, 23, 4237941811999056, 298023223876953150, 5234491418, 640550188738935, 2522015815755104284
Offset: 1
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a(n) = sumdiv(n, d, d^n*binomial(n/d, d));
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my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, ((k*x)^k)^k/(1-(k*x)^k)^(k+1)))
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from math import comb
from itertools import takewhile
from sympy import divisors
def A376015(n): return sum(d**n*comb(n//d,d) for d in takewhile(lambda d:d**2<=n,divisors(n))) # Chai Wah Wu, Sep 06 2024
A360759
a(n) = Sum_{d|n} d^(d+n/d) * binomial(d,n/d).
Original entry on oeis.org
1, 16, 243, 4112, 78125, 1680345, 40353607, 1073766400, 31381060338, 1000000781250, 34522712143931, 1283918489808640, 51185893014090757, 2177953338656796883, 98526125335697265625, 4722366482899710050304, 239072435685151324847153
Offset: 1
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a[n_] := DivisorSum[n, #^(# + n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)
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a(n) = sumdiv(n, d, d^(d+n/d)*binomial(d, n/d));
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^k*((1+k*x^k)^k-1)))
Showing 1-4 of 4 results.