A392998 - OEIS

%I #16 Jan 30 2026 23:43:31

%S 1,1,3,22,269,4236,81127,1866754,50579769,1575652024,55388271851,

%T 2168568397854,93622432268917,4419522649542196,226470310516666959,

%U 12519842071725034666,742733552305255296113,47066599327470240057456,3173076031207753527462739

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

%H Vincenzo Librandi, <a href="/A392998/b392998.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)) - (x*A(x))^3).

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

%t Table[n!*Sum[(n+k+1)^(n-3*k-1)*Binomial[n+k+1,k]/(n-3*k)!,{k,0,Floor[n/3]}],{n,0,18}] (* _Vincenzo Librandi_, Jan 30 2026 *)

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

%o (Magma) [Factorial(n) * &+[(n+k+1)^(n-3*k-1)* Binomial(n+k+1, k) / Factorial(n-3*k) : k in [0..Floor(n/3)]] : n in [0..25] ]; // _Vincenzo Librandi_, Jan 30 2026

%Y Cf. A377890.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 30 2026