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a(n) = n! * Sum_{k=0..n} k^(2*(n-k))/k!.
3

%I #9 Aug 22 2022 14:28:48

%S 1,1,3,19,253,5661,188191,8983423,594848409,52174034713,5852229698971,

%T 822684190381131,142739480367287893,30074750245383836149,

%U 7575373641076070706423,2252600759590927171373431,783103569459739402827046321,315587346190678252431713684913

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

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

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

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

%Y Cf. A354436, A356673.

%Y Cf. A234568, A356628.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 22 2022