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a(n) = n! * Sum_{k=0..floor(n/2)} (-k/2)^k / (k! * (n-2*k)!).
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%I #11 Apr 18 2023 08:28:13

%S 1,1,0,-2,7,51,-239,-2435,16353,209377,-1826099,-28232379,303020125,

%T 5494172893,-70032035163,-1457369472299,21512472563281,

%U 505400696581905,-8478758871011807,-221971772323923263,4171251104170567101,120416449897739144941

%N a(n) = n! * Sum_{k=0..floor(n/2)} (-k/2)^k / (k! * (n-2*k)!).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: exp(x) / (1 + LambertW(x^2/2)).

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

%Y Cf. A069856, A362341, A362342.

%Y Cf. A362276.

%K sign

%O 0,4

%A _Seiichi Manyama_, Apr 17 2023