A392998 Expansion of e.g.f. (1/x) * Series_Reversion( x * (exp(-x) - x^3) ).

1, 1, 3, 22, 269, 4236, 81127, 1866754, 50579769, 1575652024, 55388271851, 2168568397854, 93622432268917, 4419522649542196, 226470310516666959, 12519842071725034666, 742733552305255296113, 47066599327470240057456, 3173076031207753527462739

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

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

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A377890.

Sequence in context: A242794 A367181 A005264 * A195512 A052892 A155806

Adjacent sequences: A392995 A392996 A392997 * A392999 A393000 A393001

Keyword

nonn

Author

Seiichi Manyama, Jan 30 2026

Status

approved