A367156 - OEIS

%I #11 Nov 07 2023 08:22:59

%S 1,1,5,53,862,19024,531520,17991630,715803832,32740331784,

%T 1692869465304,97648275936672,6216826224534624,433030023365176704,

%U 32757854472395131776,2674517780432621462640,234408432378333868580736,21951787708820941049727360

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

%F a(n) = Sum_{k=0..n} (n+2*k)!/(n+k+1)! * Stirling1(n,k).

%F a(n) ~ s^2 * sqrt((2-s) / (-2*s^3 + 5*s^2 + 4*s - 4)) * n^(n-1) / (r^n * exp(n)), where r = 0.1660717422585514666099422406611296365893647754849... and s = 1.527702505127565301209742745041094767065375131037... are real roots of the system of equations 1 + s^2*log(1 + r*s) = s, 2/s - r*s^2/(1 + r*s) = 1. - _Vaclav Kotesovec_, Nov 07 2023

%t Table[Sum[(n + 2*k)!/(n + k + 1)!*StirlingS1[n, k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 07 2023 *)

%o (PARI) a(n) = sum(k=0, n, (n+2*k)!/(n+k+1)!*stirling(n, k, 1));

%Y Cf. A367155, A367157.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 07 2023