A349556 - OEIS

%I #13 Nov 22 2021 08:09:39

%S 1,1,6,69,1196,27900,820554,29168048,1216826120,58301363808,

%T 3155539049040,190434409300872,12679792851087768,923409652630222680,

%U 73016802381788896344,6230201355664856039640,570574779781503603910464,55826084651771645745562368

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

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

%F a(n) ~ s^2 * sqrt((1 - r*s) / (1 + r*s*(s-1) * (2 - r*s))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.1591040778917510493879632960549533860431737829556... and s = 1.588466710327904339474066925768589168215650366378... are real roots of the system of equations 1/s = (1 - r*s)^s, r*s/(1 - r*s) - log(1 - r*s) = 1/s. - _Vaclav Kotesovec_, Nov 22 2021

%t a[n_] := Sum[(n + k + 1)^(k - 1) * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Nov 22 2021 *)

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

%Y Cf. A052813, A162655, A349559.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 21 2021