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A343361 Expansion of Product_{k>=1} (1 + x^k)^(4^(k-1)).

%I A343361 #10 Apr 13 2021 04:43:50
%S A343361 1,1,4,20,86,390,1724,7644,33697,148401,651584,2855840,12491276,
%T A343361 54540636,237733768,1034610232,4495832776,19508749928,84540638312,
%U A343361 365888222552,1581630245756,6829047398156,29453496620000,126898489491904,546183557447366,2348560270762006,10089340886428928
%N A343361 Expansion of Product_{k>=1} (1 + x^k)^(4^(k-1)).
%F A343361 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 / (j * (4^(j-1) - 1)). - _Vaclav Kotesovec_, Apr 13 2021
%p A343361 h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A343361       add(h(n-i*j, i-1)*binomial(4^(i-1), j), j=0..n/i)))
%p A343361     end:
%p A343361 a:= n-> h(n$2):
%p A343361 seq(a(n), n=0..26);  # _Alois P. Heinz_, Apr 12 2021
%t A343361 nmax = 26; CoefficientList[Series[Product[(1 + x^k)^(4^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
%t A343361 a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 4^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]
%o A343361 (PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(4^(k-1))))} \\ _Andrew Howroyd_, Apr 12 2021
%Y A343361 Cf. A098407, A292838, A343349, A343360, A343362, A343363, A343364, A343365, A343366.
%K A343361 nonn
%O A343361 0,3
%A A343361 _Ilya Gutkovskiy_, Apr 12 2021