A392997 Expansion of e.g.f. (1/x) * Series_Reversion( x * (exp(-x) - x^2) ).

1, 1, 5, 46, 629, 11516, 264967, 7353802, 239207369, 8928133336, 376180486091, 17663931715574, 914751037702669, 51796450392492196, 3183728647902158735, 211123152550900375426, 15024331471142190589073, 1142112213489795318697904, 92366879672309716539054739

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))^2).

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

a(n) ~ (1-s)^(n+1) * n^(n-1) / (sqrt(3*(2-s^2)) * exp(n) * s^(3*n+2) * (2+s)^(n + 1/2)), where s = 0.3772780694472368061779151526950759272644649184306... is the root of the equation s + 3*exp(s)*s^2 = 1. - Vaclav Kotesovec, Jan 31 2026

Mathematica

Table[n!*Sum[(n+k+1)^(n-2*k-1)*Binomial[n+k+1, 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+1, k)/(n-2*k)!);
(Magma) [n eq 0 select 1 else Factorial(n) * &+[ Binomial(n+k+1, k) * (n+k+1)^(n-2*k-1) / Factorial(n-2*k) : k in [0..n div 2] ]: n in [0..18] ]; // Vincenzo Librandi, Jan 30 2026

Crossrefs

Cf. A377890.

Sequence in context: A121631 A071214 A052873 * A052894 A363355 A386645

Adjacent sequences: A392994 A392995 A392996 * A392998 A392999 A393000

Keyword

nonn

Author

Seiichi Manyama, Jan 30 2026

Status

approved