cp's OEIS Frontend

A319918 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^k-1).

%I A319918 #14 Dec 28 2023 07:09:25
%S A319918 1,1,4,11,32,84,230,597,1567,4020,10286,25994,65387,163065,404617,
%T A319918 997687,2448220,5977334,14530835,35173496,84814982,203760809,
%U A319918 487845377,1164191563,2769721073,6570218773,15542642042,36671354125,86306246887,202637312099,474684979292,1109539437382
%N A319918 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^k-1).
%C A319918 Convolution of A010815 and A034899.
%C A319918 Euler transform of A000225.
%H A319918 Alois P. Heinz, <a href="/A319918/b319918.txt">Table of n, a(n) for n = 0..3171</a>
%H A319918 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A319918 G.f.: exp(Sum_{k>=1} x^k/(k*(1 - x^k)*(1 - 2*x^k))).
%F A319918 a(n) ~ A247003^2 * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A065446 * sqrt(Pi) * n^(3/4)). - _Vaclav Kotesovec_, Sep 15 2021
%p A319918 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p A319918        d*(2^d-1), d=numtheory[divisors](j)), j=1..n)/n)
%p A319918     end:
%p A319918 seq(a(n), n=0..35);  # _Alois P. Heinz_, Aug 13 2021
%t A319918 nmax = 31; CoefficientList[Series[Product[1/(1 - x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
%t A319918 nmax = 31; CoefficientList[Series[Exp[Sum[x^k/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
%t A319918 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]
%Y A319918 Cf. A000225, A007070, A010815, A034691, A034899, A319919.
%K A319918 nonn
%O A319918 0,3
%A A319918 _Ilya Gutkovskiy_, Oct 01 2018