A392789 - OEIS

%I #18 Jan 24 2026 09:38:08

%S 1,1,4,30,348,5460,108000,2578800,72189600,2319166080,84119968800,

%T 3400710112800,151635740256000,7393393739571840,391334037586652160,

%U 22347717728195059200,1369596926501682278400,89664110210432082124800,6245228229359073860505600

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

%H Vincenzo Librandi, <a href="/A392789/b392789.txt">Table of n, a(n) for n = 0..300</a>

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

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

%t Table[(1/(n+1))*Sum[(2*n-3*k)!*Abs[StirlingS2[n-2*k,n-3*k]/(n-2*k)!],{k,0,Floor[n/3]}],{n,0,21}] (* _Vincenzo Librandi_, Jan 24 2026 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x+(1-exp(x^3))/x)/x))

%o (Magma) [ 1/(n+1)* &+[ Factorial(2*n-3*k)*StirlingSecond(n-2*k,n-3*k)/Factorial(n-2*k) : k in [0..Floor(n/3)] ] : n in [0..18] ]; // _Vincenzo Librandi_, Jan 24 2026

%Y Cf. A052894, A392788, A392790.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 22 2026