A355766 - OEIS

%I #14 Jul 21 2022 03:51:23

%S 1,1,8,126,3028,98540,4056948,202301456,11855415920,798682318848,

%T 60823290655680,5167260183157248,484519323081722784,

%U 49705696509114472320,5537956421036240838336,665926312161296782156800,85960998514145805006711552

%N E.g.f. satisfies A(x) = 1/(1 - x*A(x))^(A(x)^2).

%H Vaclav Kotesovec, <a href="/A355766/b355766.txt">Table of n, a(n) for n = 0..328</a>

%F a(n) = Sum_{k=0..n} (n+2*k+1)^(k-1) * |Stirling1(n,k)|.

%F a(n) ~ s^(5/2) * n^(n-1) * sqrt((1 - r*s)/(2 - 4*r*s + 2*r^2*s^2 + 3*r*s^3 - 2*r^2*s^4)) / (exp(n) * r^(n - 1/2)), where r = 0.11275303067590951818975824... and s = 1.382171434168172073998532... are real roots of the system of equations (1 - r*s)^(s^2) = 1/s, r*s/(1 - r*s) = 1/s^2 + 2*log(1 - r*s). - _Vaclav Kotesovec_, Jul 21 2022

%t Table[Sum[(n + 2*k + 1)^(k - 1)* Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 21 2022 *)

%o (PARI) a(n) = sum(k=0, n, (n+2*k+1)^(k-1)*abs(stirling(n, k, 1)));

%Y Cf. A001761, A349556.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jul 16 2022