A371117 E.g.f. satisfies A(x) = 1 - x*log(1 - x*A(x)).

1, 0, 2, 3, 32, 210, 2184, 26460, 373344, 6150816, 113958720, 2362345920, 54094694400, 1355708296800, 36926213869440, 1085886303989760, 34291129916574720, 1157362522046277120, 41576054625791078400, 1583864892141097098240, 63779322541075124428800

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)} |Stirling1(n-k,k)|/(n-2*k+1)!.

a(n) ~ sqrt(2 - r*(2*r+1)) * n^(n-1) / (exp(n) * r^n), where r = 0.4599065470184992266076522060382204730855199647380... is the root of the equation 1/r + 2*r*log(r) = 1+r. - Vaclav Kotesovec, Mar 11 2024

Mathematica

nmax = 20; A[_] = 0; Do[A[x_] = 1 - x*Log[1 - x*A[x]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 11 2024 *)
Table[n!*Sum[Abs[StirlingS1[n- k, k]]/(n-2 k+1)!, {k, 0, Floor[n/2]}], {n, 0, 25}] (* Vincenzo Librandi, Feb 22 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/(n-2*k+1)!);
(Magma) [Factorial(n)* &+[Abs(StirlingFirst(n-k, k)) /Factorial(n-2*k+1): k in [0..Floor(n/2)] ] : n in [0..23] ]; // Vincenzo Librandi, Feb 22 2026

Crossrefs

Cf. A138013, A371118.

Cf. A371115.

Sequence in context: A357265 A356904 A041895 * A277481 A079883 A358041

Adjacent sequences: A371114 A371115 A371116 * A371118 A371119 A371120

Keyword

nonn

Author

Seiichi Manyama, Mar 11 2024

Status

approved