A392886 - OEIS

%I #14 Feb 05 2026 09:26:05

%S 1,1,2,6,36,420,6120,94920,1599360,30844800,694008000,17786260800,

%T 502352928000,15367975983360,507522467612160,18092546664192000,

%U 694083913915392000,28487571264011059200,1243715592543834009600,57518280664660995379200,2809850302506602234880000

%N Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 - log(1-x^3)/x^2) ).

%H Vincenzo Librandi, <a href="/A392886/b392886.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f. A(x) satisfies A(x) = 1 - log(1-(x*A(x))^3)/(x*A(x))^2.

%F a(n) = (n!)^2 * Sum_{k=0..floor(n/3)} 1/(3*k+1)! * |Stirling1(n-2*k,n-3*k)|/(n-2*k)!.

%t Table[(n!)^2* Sum[1/(3*k+1)!*Abs[StirlingS1[n-2*k,n-3*k]/(n-2*k)!],{k,0,Floor[n/3]}],{n,0,20}] (* _Vincenzo Librandi_, Feb 05 2026 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1-log(1-x^3)/x^2))/x))

%o (Magma) [Factorial(n)^2 * &+[1/Factorial(3*k+1) * Abs(StirlingFirst(n - 2*k, n-3*k) / Factorial(n - 2*k)): k in [0..Floor(n/3)]]: n in [0..25] ]; // _Vincenzo Librandi_, Feb 05 2026

%Y Cf. A138013, A391838.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 25 2026