A252782 - OEIS

%I #20 Mar 22 2017 12:21:17

%S 1,1,5,36,490,12729,689896,70223666,13803604854,5567490203192,

%T 4386006155453382,6711625359213752077,21048250447828058144403,

%U 131214686495783317936950378,1603891839732647136012816743764,40296598014204065945778862754895836

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

%H Alois P. Heinz, <a href="/A252782/b252782.txt">Table of n, a(n) for n = 0..80</a>

%H Vaclav Kotesovec, <a href="/A252782/a252782.jpg">Graph - The asymptotic ratio</a>

%F a(n) = [x^n] Product_{j>=1} 1/(1-x^j)^(j^n).

%F Conjecture: limit n->infinity a(n)^(1/n^2) = exp(exp(-1)) = 1.444667861... . - _Vaclav Kotesovec_, Mar 25 2016

%p with(numtheory):

%p A:= proc(n, k) option remember; `if`(n=0, 1, add(add(

%p        d*d^k, d=divisors(j))*A(n-j, k), j=1..n)/n)

%p     end:

%p a:= n-> A(n$2):

%p seq(a(n), n=0..20);

%t Table[SeriesCoefficient[Product[1/(1-x^k)^(k^n),{k,1,n}],{x,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 01 2015 *)

%Y Main diagonal of A144048.

%Y Cf. A008485, A073229, A255672, A270917.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Dec 21 2014