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A343360 Expansion of Product_{k>=1} (1 + x^k)^(3^(k-1)).

%I A343360 #10 Apr 13 2021 04:43:24
%S A343360 1,1,3,12,39,138,469,1603,5427,18372,61869,207909,696537,2328039,
%T A343360 7762266,25826142,85749969,284171598,940027872,3104280885,10234808334,
%U A343360 33692547249,110753171784,363561071175,1191860487561,3902350627434,12761565487173,41685086306917,136012008938158
%N A343360 Expansion of Product_{k>=1} (1 + x^k)^(3^(k-1)).
%F A343360 a(n) ~ exp(2*sqrt(n/3) - 1/6 - c/3) * 3^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (3^(j-1) - 1)). - _Vaclav Kotesovec_, Apr 13 2021
%p A343360 h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A343360       add(h(n-i*j, i-1)*binomial(3^(i-1), j), j=0..n/i)))
%p A343360     end:
%p A343360 a:= n-> h(n$2):
%p A343360 seq(a(n), n=0..28);  # _Alois P. Heinz_, Apr 12 2021
%t A343360 nmax = 28; CoefficientList[Series[Product[(1 + x^k)^(3^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
%t A343360 a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 3^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 28}]
%o A343360 (PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(3^(k-1))))} \\ _Andrew Howroyd_, Apr 12 2021
%Y A343360 Cf. A098407, A104460, A256142, A343361, A343362, A343363, A343364, A343365, A343366.
%K A343360 nonn
%O A343360 0,3
%A A343360 _Ilya Gutkovskiy_, Apr 12 2021