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

%I #39 Sep 16 2022 12:13:56

%S 1,1,4,33,448,8105,192576,5946913,226097152,10389920913,571788928000,

%T 36818407010561,2741300619657216,234014330510734969,

%U 22620660476040331264,2457467449742570271105,298061856229112792743936,40058727579693211737837857

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

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

%p f:= proc(n) local k; n! * add((n-2*k)^n/(n-2*k)!,k=0..floor(n/2)) end proc:

%p map(f, [$0..20]); # _Robert Israel_, Sep 16 2022

%t a[n_] := n! * Sum[(n - 2*k)^n/(n - 2*k)!, {k, 0, Floor[n/2]} ]; a[0] = 1; Array[a, 18, 0] (* _Amiram Eldar_, Sep 16 2022 *)

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

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

%Y Cf. A256016, A357174.

%Y Cf. A352082, A357146.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Sep 16 2022