A392999 Expansion of e.g.f. (1/x) * Series_Reversion( x/(exp(x) + x^2) ).

1, 1, 5, 34, 365, 4996, 86287, 1779478, 42935993, 1184939272, 36859566491, 1275944729434, 48663707221285, 2027692958251468, 91655762781872759, 4467378174077738686, 233570770252108053233, 13040161327045258238992, 774302485511519895259315

Offset

0,3

Links

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

Formula

E.g.f. A(x) satisfies A(x) = exp(x*A(x)) + (x*A(x))^2.

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

Mathematica

Table[n!*Sum[(n-k+1)^(n-2*k-1)*Binomial[n, k]/(n-2*k)!, {k, 0, Floor[n/2]}], {n, 0, 18}] (* Vincenzo Librandi, Jan 30 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, (n-k+1)^(n-2*k-1)*binomial(n, k)/(n-2*k)!);
(Magma) [Factorial(n) * &+[(n-k+1)^(n-2*k-1)* Binomial(n, k) / Factorial(n-2*k) : k in [0..Floor(n/2)]] : n in [0..25] ]; // Vincenzo Librandi, Jan 30 2026

Crossrefs

Cf. A352410.

Sequence in context: A133297 A054931 A362771 * A276753 A226554 A211037

Adjacent sequences: A392996 A392997 A392998 * A393000 A393001 A393002

Keyword

nonn

Author

Seiichi Manyama, Jan 30 2026

Status

approved