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A343362 Expansion of Product_{k>=1} (1 + x^k)^(5^(k-1)).

%I A343362 #10 Apr 13 2021 04:44:15
%S A343362 1,1,5,30,160,885,4810,26185,142005,769305,4159301,22455876,121057525,
%T A343362 651737675,3504241650,18818709130,100945053055,540885242825,
%U A343362 2895159035375,15481318817450,82704855762375,441427664993275,2354020475714775,12542918682786300,66778882780674975
%N A343362 Expansion of Product_{k>=1} (1 + x^k)^(5^(k-1)).
%F A343362 a(n) ~ exp(2*sqrt(n/5) - 1/10 - c/5) * 5^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (5^(j-1) - 1)). - _Vaclav Kotesovec_, Apr 13 2021
%p A343362 h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A343362       add(h(n-i*j, i-1)*binomial(5^(i-1), j), j=0..n/i)))
%p A343362     end:
%p A343362 a:= n-> h(n$2):
%p A343362 seq(a(n), n=0..24);  # _Alois P. Heinz_, Apr 12 2021
%t A343362 nmax = 24; CoefficientList[Series[Product[(1 + x^k)^(5^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
%t A343362 a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 5^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 24}]
%o A343362 (PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(5^(k-1))))} \\ _Andrew Howroyd_, Apr 12 2021
%Y A343362 Cf. A098407, A292839, A343350, A343360, A343361, A343363, A343364, A343365, A343366.
%K A343362 nonn
%O A343362 0,3
%A A343362 _Ilya Gutkovskiy_, Apr 12 2021