A319110 Expansion of Product_{k>=1} 1/(1 - (k - 1)*x^k).

1, 0, 1, 2, 4, 6, 13, 18, 37, 56, 101, 152, 285, 410, 713, 1118, 1830, 2780, 4618, 6934, 11278, 17092, 26894, 40822, 64435, 96372, 149299, 225104, 345131, 515394, 788176, 1169962, 1772957, 2632458, 3950365, 5849260, 8748993, 12867848, 19135894, 28126614, 41598695

Offset: 0

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..8190
• Vaclav Kotesovec, Closed form of the asymptotics

Crossrefs

Cf. A006906, A034008, A052847, A074141, A267007.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1,
       0^n, b(n, i-1)+(i-1)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 19 2019

Mathematica

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

Formula

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

From Vaclav Kotesovec, Sep 11 2018: (Start)

a(n) ~ c * 2^(2*n/5), where

c = 28108804.248904780960402246466460350520790117596512766842168... if mod(n,5) = 0

c = 28108804.010850549080284030388905319123062152339902207992657... if mod(n,5) = 1

c = 28108804.067769166625741650205643600577757560110636366636106... if mod(n,5) = 2

c = 28108804.083581827971851596540314974909801290757084687583764... if mod(n,5) = 3

c = 28108804.058853893104368046896759214442695016905368229405793... if mod(n,5) = 4

(End)