A316961 Expansion of Product_{k>=1} 1/(1 - sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

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

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..4150

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

Cf. A000203, A061256, A180305, A279784, A316962.

Sequence in context: A062015 A006640 A212686 * A180002 A266821 A306484

Adjacent sequences: A316958 A316959 A316960 * A316962 A316963 A316964

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 17 2018

Status

approved