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A270917 Coefficient of x^n in Product_{k>=1} (1 + x^k)^(k^n).

%I A270917 #15 Oct 16 2017 10:47:48
%S A270917 1,1,4,35,457,12421,678101,69540142,13730026114,5551573311817,
%T A270917 4379029522335786,6705866900012021577,21038900445652125741759,
%U A270917 131183458646068931932668114,1603688863449847489871671547959,40294004792352613617780682256221711
%N A270917 Coefficient of x^n in Product_{k>=1} (1 + x^k)^(k^n).
%H A270917 Vaclav Kotesovec, <a href="/A270917/b270917.txt">Table of n, a(n) for n = 0..80</a>
%F A270917 Conjecture: limit n->infinity a(n)^(1/n^2) = exp(exp(-1)) = 1.444667861...
%p A270917 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A270917       add(b(n-i*j, i-1, k)*binomial(i^k, j), j=0..n/i)))
%p A270917     end:
%p A270917 a:= n-> b(n$3):
%p A270917 seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 16 2017
%t A270917 Table[SeriesCoefficient[Product[(1+x^k)^(k^n), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
%Y A270917 Cf. A252782, A255672, A270913.
%Y A270917 Main diagonal of A284992.
%K A270917 nonn
%O A270917 0,3
%A A270917 _Vaclav Kotesovec_, Mar 25 2016