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

%I A319110 #14 Dec 28 2023 07:09:18
%S A319110 1,0,1,2,4,6,13,18,37,56,101,152,285,410,713,1118,1830,2780,4618,6934,
%T A319110 11278,17092,26894,40822,64435,96372,149299,225104,345131,515394,
%U A319110 788176,1169962,1772957,2632458,3950365,5849260,8748993,12867848,19135894,28126614,41598695
%N A319110 Expansion of Product_{k>=1} 1/(1 - (k - 1)*x^k).
%H A319110 Vaclav Kotesovec, <a href="/A319110/b319110.txt">Table of n, a(n) for n = 0..8190</a>
%H A319110 Vaclav Kotesovec, <a href="/A319110/a319110.jpg">Closed form of the asymptotics</a>
%F A319110 G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j - 1)^k*x^(j*k)/k).
%F A319110 From _Vaclav Kotesovec_, Sep 11 2018: (Start)
%F A319110 a(n) ~ c * 2^(2*n/5), where
%F A319110 c = 28108804.248904780960402246466460350520790117596512766842168... if mod(n,5) = 0
%F A319110 c = 28108804.010850549080284030388905319123062152339902207992657... if mod(n,5) = 1
%F A319110 c = 28108804.067769166625741650205643600577757560110636366636106... if mod(n,5) = 2
%F A319110 c = 28108804.083581827971851596540314974909801290757084687583764... if mod(n,5) = 3
%F A319110 c = 28108804.058853893104368046896759214442695016905368229405793... if mod(n,5) = 4
%F A319110 (End)
%p A319110 b:= proc(n, i) option remember; `if`(n=0 or i=1,
%p A319110        0^n, b(n, i-1)+(i-1)*b(n-i, min(n-i, i)))
%p A319110     end:
%p A319110 a:= n-> b(n$2):
%p A319110 seq(a(n), n=0..42);  # _Alois P. Heinz_, Aug 19 2019
%t A319110 nmax = 40; CoefficientList[Series[Product[1/(1 - (k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A319110 nmax = 40; CoefficientList[Series[Exp[Sum[Sum[(j - 1)^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
%t A319110 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}]
%Y A319110 Cf. A006906, A034008, A052847, A074141, A267007.
%K A319110 nonn
%O A319110 0,4
%A A319110 _Ilya Gutkovskiy_, Sep 10 2018