A392929 E.g.f. A(x) satisfies A(x) = 1 - (1/x) * log(1 - x^2*A(x)).

1, 1, 2, 9, 60, 520, 5640, 73500, 1118880, 19501776, 383110560, 8377205760, 201831315840, 5312936154240, 151725201788160, 4672298462832000, 154342559776204800, 5444359223483980800, 204256674378018662400, 8121472917226507238400, 341153109889323443712000

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

a(n) ~ sqrt(r - 1 + 1/r) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.443039301440488027350476160766968406... is the root of the equation r - log(r) = (1-r)/r. - _Vaclav Kotesovec_, Jan 27 2026

Mathematica

Table[n!* Sum[1/(k+1)!*Abs[StirlingS1[n-k, n-2*k]], {k, 0, Floor[n/2]}], {n, 0, 20}] (* _Vincenzo Librandi_, Jan 28 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, 1/(k+1)!*abs(stirling(n-k, n-2*k, 1)));
(Magma) [Factorial(n)* &+[1/Factorial(k+1) * Abs(StirlingFirst(n-k, n-2*k)): k in [0..Floor(n/2)] ] : n in [0..23] ]; // _Vincenzo Librandi_, Jan 28 2026

Crossrefs

Cf. A392930.

Sequence in context: A111558 A322943 A168449 * A001193 A161391 A120014

Adjacent sequences: A392926 A392927 A392928 * A392930 A392931 A392932

Keyword

nonn

Author

_Seiichi Manyama_, Jan 27 2026

Status

approved