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

%I A319756 #12 Dec 06 2020 01:14:08
%S A319756 1,0,1,2,5,6,18,20,52,76,151,214,486,638,1265,1990,3572,5288,9968,
%T A319756 14568,26270,40246,68326,104414,182191,271892,457636,708012,1164554,
%U A319756 1774422,2945077,4450020,7261298,11138514,17827308,27228060,43860232,66305840,105486224,161284674,253846152
%N A319756 Expansion of Product_{k>=1} (1 - x^k)/(1 - k*x^k).
%C A319756 Convolution of A006906 and A010815.
%H A319756 Seiichi Manyama, <a href="/A319756/b319756.txt">Table of n, a(n) for n = 0..5000</a>
%F A319756 G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(d^(k/d) - 1) ) * x^k/k).
%F A319756 From _Vaclav Kotesovec_, Sep 27 2018: (Start)
%F A319756 a(n) ~ c * 3^(n/3), where
%F A319756 c = 4613.026226899587659466790384528262900057997961519... if mod(n,3)=0
%F A319756 c = 4612.491093385908314202944836907761153110706939289... if mod(n,3)=1
%F A319756 c = 4612.543916007416515763773288072302642108310934844... if mod(n,3)=2
%F A319756 (End)
%p A319756 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
%t A319756 nmax = 40; CoefficientList[Series[Product[(1 - x^k)/(1 - k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A319756 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]
%t A319756 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}]
%o A319756 (PARI) x='x+O('x^40); Vec(prod(n=1, 40, (1-x^n)/(1-n*x^n))) \\ _Altug Alkan_, Sep 27 2018
%Y A319756 Cf. A006906, A010815, A267005.
%K A319756 nonn
%O A319756 0,4
%A A319756 _Ilya Gutkovskiy_, Sep 27 2018