A371115 E.g.f. satisfies A(x) = 1 + x*(exp(x*A(x)) - 1).

1, 0, 2, 3, 28, 185, 1566, 18277, 218744, 3206961, 52134490, 935303501, 18733723812, 406458491881, 9598660337462, 244471271572725, 6671672053304176, 194631575264393057, 6036199529439919410, 198427339307102272669, 6892068588221322730460

Offset

0,3

Links

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} Stirling2(n-k,k)/(n-2*k+1)!.

From Vaclav Kotesovec, Mar 11 2024: (Start)

E.g.f.: 1 - x - LambertW(-exp((1 - x)*x)*x^2)/x.

a(n) ~ sqrt(2 + r - 2*r^2) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.5356007344755967412570670018666980389185523835846... if the root of the equation exp(1 + r - r^2) * r^2 = 1. (End)

Mathematica

nmax = 20; CoefficientList[Series[1 - x - ProductLog[-E^((1 - x)*x)*x^2]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 11 2024 *)
Table[n!*Sum[StirlingS2[n-k, k]/(n-2k+1)!, {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Feb 25 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, stirling(n-k, k, 2)/(n-2*k+1)!);
(Magma) [Factorial(n)* &+[StirlingSecond(n-k, k) /Factorial(n-2*k+1): k in [0..Floor(n/2)] ] : n in [0..23] ]; // Vincenzo Librandi, Feb 25 2026

Crossrefs

Cf. A000272, A371116.

Cf. A371117.

Sequence in context: A074233 A354611 A356906 * A206592 A126266 A219975

Adjacent sequences: A371112 A371113 A371114 * A371116 A371117 A371118

Keyword

nonn

Author

Seiichi Manyama, Mar 11 2024

Status

approved