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A252782 a(n) = n-th term of Euler transform of n-th powers.

%I A252782 #20 Mar 22 2017 12:21:17
%S A252782 1,1,5,36,490,12729,689896,70223666,13803604854,5567490203192,
%T A252782 4386006155453382,6711625359213752077,21048250447828058144403,
%U A252782 131214686495783317936950378,1603891839732647136012816743764,40296598014204065945778862754895836
%N A252782 a(n) = n-th term of Euler transform of n-th powers.
%H A252782 Alois P. Heinz, <a href="/A252782/b252782.txt">Table of n, a(n) for n = 0..80</a>
%H A252782 Vaclav Kotesovec, <a href="/A252782/a252782.jpg">Graph - The asymptotic ratio</a>
%F A252782 a(n) = [x^n] Product_{j>=1} 1/(1-x^j)^(j^n).
%F A252782 Conjecture: limit n->infinity a(n)^(1/n^2) = exp(exp(-1)) = 1.444667861... . - _Vaclav Kotesovec_, Mar 25 2016
%p A252782 with(numtheory):
%p A252782 A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
%p A252782        d*d^k, d=divisors(j))*A(n-j, k), j=1..n)/n)
%p A252782     end:
%p A252782 a:= n-> A(n$2):
%p A252782 seq(a(n), n=0..20);
%t A252782 Table[SeriesCoefficient[Product[1/(1-x^k)^(k^n),{k,1,n}],{x,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 01 2015 *)
%Y A252782 Main diagonal of A144048.
%Y A252782 Cf. A008485, A073229, A255672, A270917.
%K A252782 nonn
%O A252782 0,3
%A A252782 _Alois P. Heinz_, Dec 21 2014