A392850 - OEIS

%I #13 Jan 26 2026 09:43:02

%S 1,0,0,6,36,150,2700,54726,771876,12945750,327177180,8394923526,

%T 209893127796,6106875462870,203608601845740,7013022801220806,

%U 253941767834096196,10074938765228850390,427741076842456307580,18976040774569773250566,888872049703600193539476

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

%H Vincenzo Librandi, <a href="/A392850/b392850.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)) - 1)^3.

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

%t Table[(1/(n+1))*Sum[(3*k)!*Binomial[n+1,k]*Abs[StirlingS2[n,3*k]],{k,0,Floor[n/3]}],{n,0,23}] (* _Vincenzo Librandi_, Jan 25 2026 *)

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

%o (Magma) [(1/(n+1))* &+[Factorial(3*k)*Binomial(n+1,k)*Abs(StirlingSecond(n, 3*k)): k in [0..Floor(n/3)] ] : n in [0..23] ]; // _Vincenzo Librandi_, Jan 25 2026

%Y Cf. A392828, A392851.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Jan 25 2026