A392759 - OEIS

%I #18 Jan 26 2026 09:42:59

%S 1,0,2,6,94,940,17348,309708,7306416,184734576,5473530432,

%T 176925756480,6380745801648,249933477549600,10665118139638272,

%U 489943094795975520,24179868161674982784,1273841570260090509312,71417817462566089624320,4243564232503805507977728

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

%H Vincenzo Librandi, <a href="/A392759/b392759.txt">Table of n, a(n) for n = 0..300</a>

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

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

%t Table[(1/(n+1)!)*Sum[(2 k)!/k!*(n+k)!*Abs[StirlingS1[n,2 k]],{k,0,Floor[n/2]}],{n,0,19}] (* _Vincenzo Librandi_, Jan 23 2026 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-log(1-x)^2))/x))

%o (Magma) function a(n) s := 0; for k in [0..Floor(n/2)] do s +:= Factorial(2*k)/Factorial(k) * Factorial(n+k) * Abs(StirlingFirst(n, 2*k)); end for; return s / Factorial(n+1); end function;

%o [a(n) : n in [0..19]]; // _Vincenzo Librandi_, Jan 23 2026

%Y Cf. A392791, A392792, A392793.

%Y Cf. A052802, A143154.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 21 2026