A319756 Expansion of Product_{k>=1} (1 - x^k)/(1 - k*x^k).

1, 0, 1, 2, 5, 6, 18, 20, 52, 76, 151, 214, 486, 638, 1265, 1990, 3572, 5288, 9968, 14568, 26270, 40246, 68326, 104414, 182191, 271892, 457636, 708012, 1164554, 1774422, 2945077, 4450020, 7261298, 11138514, 17827308, 27228060, 43860232, 66305840, 105486224, 161284674, 253846152

Offset

0,4

Comments

Convolution of A006906 and A010815.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Formula

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(d^(k/d) - 1) ) * x^k/k).

From Vaclav Kotesovec, Sep 27 2018: (Start)

a(n) ~ c * 3^(n/3), where

c = 4613.026226899587659466790384528262900057997961519... if mod(n,3)=0

c = 4612.491093385908314202944836907761153110706939289... if mod(n,3)=1

c = 4612.543916007416515763773288072302642108310934844... if mod(n,3)=2

(End)

Maple

a:=series(mul((1-x^k)/(1-k*x^k), k=1..100), x=0, 41): seq(coeff(a, x, n), n=0..40); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 40; CoefficientList[Series[Product[(1 - x^k)/(1 - k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[d (d^(k/d) - 1), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (d^(k/d) - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

Prog

(PARI) x='x+O('x^40); Vec(prod(n=1, 40, (1-x^n)/(1-n*x^n))) \\ Altug Alkan, Sep 27 2018

Crossrefs

Cf. A006906, A010815, A267005.

Sequence in context: A146477 A348439 A166753 * A202854 A274911 A282536

Adjacent sequences: A319753 A319754 A319755 * A319757 A319758 A319759

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 27 2018

Status

approved