A392852 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + log(1-x)^2) ).

1, 0, 2, 6, 70, 700, 10268, 171108, 3404280, 77158800, 1977004752, 56398061280, 1774104501024, 61004101979808, 2276675340590688, 91652238159138720, 3959064663300753408, 182664424801700783616, 8965482639644918631168, 466444129035055869430272

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 + log(1-x*A(x))^2.

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (2*k)! * binomial(n+1,k) * |Stirling1(n,2*k)|.

a(n) ~ sqrt(2 - 4*sqrt(r) + 4*r) * (1 - sqrt(r)) * n^(n-1) / (exp(n) * r^(n + 3/4)), where r = 0.3380187987694888344077904092062555373374648295224... is the root of the equation 1/sqrt(r) + log(2*sqrt(r) - 2*r) = 1. - Vaclav Kotesovec, Feb 03 2026

Mathematica

Table[(1/(n+1))*Sum[(2*k)!*Binomial[n+1, k]*Abs[StirlingS1[n, 2*k]], {k, 0, Floor[n/2]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 26 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\2, (2*k)!*binomial(n+1, k)*abs(stirling(n, 2*k, 1)))/(n+1);
(Magma) [(1/(n+1))* &+[Factorial(2*k)*Binomial(n+1, k)*Abs(StirlingFirst(n, 2*k)): k in [0..Floor(n/2)] ] : n in [0..23] ]; // Vincenzo Librandi, Jan 26 2026

Crossrefs

Cf. A138013, A392853.

Cf. A392829, A392831.

Sequence in context: A097419 A219037 A156458 * A244494 A136268 A030242

Adjacent sequences: A392849 A392850 A392851 * A392853 A392854 A392855

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2026

Status

approved