A392890 - OEIS

%I #18 Jan 30 2026 09:53:29

%S 1,1,2,6,36,420,6120,94080,1545600,28667520,620676000,15462770400,

%T 427269427200,12769595879040,409975771645440,14163639025536000,

%U 526670239685990400,20988647304174950400,890661296247564441600,40028284097655549696000,1898798845826870507520000

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

%H Vincenzo Librandi, <a href="/A392890/b392890.txt">Table of n, a(n) for n = 0..300</a>

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

%F a(n) = (n!)^2 * Sum_{k=0..floor(n/3)} 1/(3*k+1)! * Stirling2(n-2*k,n-3*k)/(n-2*k)!.

%t Table[(n!)^2*Sum[1/(3*k+1)!*Abs[StirlingS2[n-2*k,n-3*k]/(n-2*k)!],{k,0,Floor[n/3]}],{n,0,23}] (* _Vincenzo Librandi_, Jan 26 2026 *)

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

%o (Magma) [ Factorial(n)^2 * &+[1 / Factorial(3*k + 1) * Abs(StirlingSecond(n - 2*k, n - 3*k) / Factorial(n - 2*k)): k in [0..Floor(n/3)]] : n in [0..25] ]; // _Vincenzo Librandi_, Jan 26 2026

%Y Cf. A000272, A392889.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 25 2026