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A316961 Expansion of Product_{k>=1} 1/(1 - sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203). 3
1, 1, 4, 8, 24, 42, 118, 208, 524, 961, 2191, 3994, 9020, 16142, 34500, 62814, 130496, 234474, 478334, 855982, 1712012, 3061230, 6003546, 10689178, 20783796, 36789875, 70540531, 124812892, 237022708, 417422168, 786509778, 1381137702, 2583046168, 4526024200, 8402928681 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
G.f.: exp(Sum_{k>=1} Sum_{j>=1} sigma(j)^k*x^(j*k)/k).
From Vaclav Kotesovec, Jul 28 2018: (Start)
a(n) ~ c * 3^(n/2), where
c = 133.83151651318934683776776253692818185240361972305... if n is even and
c = 131.63961163168586786976253326691345807212512512772... if n is odd.
In closed form, a(n) ~ ((3 + sqrt(3)) * Product_{k>=3} (1/(1 - sigma(k) / 3^(k/2))) + (-1)^n * (3 - sqrt(3)) * Product_{k>=3} (1/(1 - (-1)^k * sigma(k) / 3^(k/2)))) * 3^(n/2) / 4. (End)
MAPLE
with(numtheory): a:=series(mul(1/(1-sigma(k)*x^k), k=1..100), x=0, 35): seq(coeff(a, x, n), n=0..34); # Paolo P. Lava, Apr 02 2019
MATHEMATICA
nmax = 34; CoefficientList[Series[Product[1/(1 - DivisorSigma[1, k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 34; CoefficientList[Series[Exp[Sum[Sum[DivisorSigma[1, j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DivisorSigma[1, d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]
CROSSREFS
Sequence in context: A062015 A006640 A212686 * A180002 A266821 A306484
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jul 17 2018
STATUS
approved

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Last modified February 25 16:57 EST 2024. Contains 370332 sequences. (Running on oeis4.)