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A343349 Expansion of Product_{k>=1} 1 / (1 - x^k)^(4^(k-1)).

%I A343349 #9 Apr 12 2021 12:12:51
%S A343349 1,1,5,21,95,415,1851,8155,36030,158510,696502,3052966,13359230,
%T A343349 58346206,254405630,1107479694,4813850699,20894227355,90567536543,
%U A343349 392066476815,1695180397145,7320927664713,31581573600685,136094434672509,585876330191950,2519701493092958
%N A343349 Expansion of Product_{k>=1} 1 / (1 - x^k)^(4^(k-1)).
%F A343349 a(n) ~ exp(sqrt(n) - 1/8 + c/4) * 2^(2*n - 3/2) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (4^(j-1) - 1)). - _Vaclav Kotesovec_, Apr 12 2021
%p A343349 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p A343349       d*4^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
%p A343349     end:
%p A343349 seq(a(n), n=0..25);  # _Alois P. Heinz_, Apr 12 2021
%t A343349 nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^(4^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
%t A343349 a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]
%Y A343349 Cf. A034691, A104460, A144068, A343350, A343351, A343352, A343353, A343354, A343355.
%K A343349 nonn
%O A343349 0,3
%A A343349 _Ilya Gutkovskiy_, Apr 12 2021