A392852 - OEIS

%I #16 Feb 03 2026 03:39:23

%S 1,0,2,6,70,700,10268,171108,3404280,77158800,1977004752,56398061280,

%T 1774104501024,61004101979808,2276675340590688,91652238159138720,

%U 3959064663300753408,182664424801700783616,8965482639644918631168,466444129035055869430272

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

%H Vincenzo Librandi, <a href="/A392852/b392852.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f. A(x) satisfies A(x) = 1 + log(1-x*A(x))^2.

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

%F 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

%t 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 *)

%o (PARI) a(n) = sum(k=0, n\2, (2*k)!*binomial(n+1, k)*abs(stirling(n, 2*k, 1)))/(n+1);

%o (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

%Y Cf. A138013, A392853.

%Y Cf. A392829, A392831.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 25 2026