A367156 E.g.f. satisfies A(x) = 1 + A(x)^2 * log(1 + x*A(x)).

1, 1, 5, 53, 862, 19024, 531520, 17991630, 715803832, 32740331784, 1692869465304, 97648275936672, 6216826224534624, 433030023365176704, 32757854472395131776, 2674517780432621462640, 234408432378333868580736, 21951787708820941049727360

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

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

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

Mathematica

Table[Sum[(n + 2*k)!/(n + k + 1)!*StirlingS1[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 07 2023 *)

Prog

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

Crossrefs

Cf. A367155, A367157.

Sequence in context: A036916 A118583 A090360 * A123130 A094089 A357343

Adjacent sequences: A367153 A367154 A367155 * A367157 A367158 A367159

Keyword

nonn

Author

Seiichi Manyama, Nov 07 2023

Status

approved