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A343365 Expansion of Product_{k>=1} (1 + x^k)^(8^(k-1)).

%I A343365 #10 Apr 13 2021 04:45:33
%S A343365 1,1,8,72,604,5148,43544,368408,3112262,26273542,221605240,1867736120,
%T A343365 15730022540,132385106956,1113413229000,9358220560136,78606905495809,
%U A343365 659886123312449,5536404584185376,46424396382193376,389074608184431328,3259085506224931424,27286163457927575200
%N A343365 Expansion of Product_{k>=1} (1 + x^k)^(8^(k-1)).
%F A343365 a(n) ~ exp(sqrt(n/2) - 1/16 - c/8) * 2^(3*n - 7/4) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (8^(j-1) - 1)). - _Vaclav Kotesovec_, Apr 13 2021
%p A343365 h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A343365       add(h(n-i*j, i-1)*binomial(8^(i-1), j), j=0..n/i)))
%p A343365     end:
%p A343365 a:= n-> h(n$2):
%p A343365 seq(a(n), n=0..22);  # _Alois P. Heinz_, Apr 12 2021
%t A343365 nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(8^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
%t A343365 a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 8^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]
%o A343365 (PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(8^(k-1))))} \\ _Andrew Howroyd_, Apr 12 2021
%Y A343365 Cf. A098407, A292842, A343353, A343360, A343361, A343362, A343363, A343364, A343366.
%K A343365 nonn
%O A343365 0,3
%A A343365 _Ilya Gutkovskiy_, Apr 12 2021