A392832 - OEIS

%I #19 Feb 18 2026 08:03:00

%S 1,0,0,0,0,120,1080,8820,75600,701568,25235280,736589040,17454333600,

%T 376218063936,7825958011872,207779505635520,7160585771888640,

%U 266170809822432768,9680346223607877120,339097779148341335040,12234333214554427699200,488121623700513688166400

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

%H Vincenzo Librandi, <a href="/A392832/b392832.txt">Table of n, a(n) for n = 0..300</a>

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

%F a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/5)} (3*k)! * binomial(n+1,k) * |Stirling1(n-2*k,3*k)|/(n-2*k)!.

%t Table[(n!/(n+1))* Sum[(3*k)!*Binomial[n+1,k]*Abs[StirlingS1[n-2*k,3*k]]/(n-2*k)!,{k,0,Floor[n/5]}],{n,0,21}] (* _Vincenzo Librandi_, Feb 07 2026 *)

%o (PARI) a(n) = n!*sum(k=0, n\5, (3*k)!*binomial(n+1, k)*abs(stirling(n-2*k, 3*k, 1))/(n-2*k)!)/(n+1);

%o (Magma) [(Factorial (n)/(n+1)) * &+[Factorial(3*k)* Binomial(n+1,k)* Abs(StirlingFirst(n - 2*k, 3*k)) / Factorial(n -2*k): k in [0..Floor(n/5)]]: n in [0..25] ]; // _Vincenzo Librandi_, Feb 07 2026

%Y Cf. A371138, A392831.

%Y Cf. A392828.

%K nonn

%O 0,6

%A _Seiichi Manyama_, Jan 24 2026