A392929 - OEIS

%I #16 Jan 28 2026 08:57:38

%S 1,1,2,9,60,520,5640,73500,1118880,19501776,383110560,8377205760,

%T 201831315840,5312936154240,151725201788160,4672298462832000,

%U 154342559776204800,5444359223483980800,204256674378018662400,8121472917226507238400,341153109889323443712000

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

%H Vincenzo Librandi, <a href="/A392929/b392929.txt">Table of n, a(n) for n = 0..300</a>

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

%F 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

%t 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 *)

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

%o (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

%Y Cf. A392930.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 27 2026