A392765 - OEIS

%I #12 Feb 13 2026 12:30:41

%S 1,0,2,6,86,870,14942,264726,5933174,146722470,4174735982,

%T 131077743126,4560210165062,173031374724390,7134885213734462,

%U 317190775386859926,15136241602198757270,771407543960187434790,41827159120486727660942,2404019686442874374551446

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

%H Vincenzo Librandi, <a href="/A392765/b392765.txt">Table of n, a(n) for n = 0..300</a>

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

%F a(n) = (1/(n+1)!) * Sum_{k=0..floor(n/2)} (2*k)!/k! * (n+k)! * Stirling2(n,2*k).

%t Table[(1/(n+1)!)* Sum[(2*k)!/k!*(n+k)!*StirlingS2[n,2*k],{k,0,Floor[n/2]}],{n,0,21}] (* _Vincenzo Librandi_, Feb 13 2026 *)

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

%o (Magma) [(1/Factorial(n+1)) * &+[Factorial(2*k) / Factorial(k) * Factorial(n+k)* StirlingSecond(n, 2*k): k in [0..Floor(n/2)]]: n in [0..25] ]; // _Vincenzo Librandi_, Feb 13 2026

%Y Cf. A052894.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 22 2026