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

%I #9 Sep 17 2022 08:44:53

%S 1,0,2,6,216,2040,111240,2164680,159391680,5247305280,491431600800,

%T 24437592194400,2800955712804480,195393943295591040,

%U 26699221909806526080,2479967110139382864000,396503602252401676032000,47167550656581451928832000

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

%F E.g.f.: Sum_{k>=0} (k * x)^(2 * k) / (k! * (1 - k * x)).

%o (PARI) a(n) = n!*sum(k=0, n\2, k^n/k!);

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^(2*k)/(k!*(1-k*x)))))

%Y Cf. A256016, A357192.

%Y Cf. A352981, A357193.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Sep 17 2022