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a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(n-k).
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%I #15 Apr 16 2022 09:39:23

%S 1,1,4,28,264,3207,47696,839412,17061280,393264145,10135913792,

%T 288839201432,9017184333440,306045200463519,11220008681600256,

%U 441866073895351128,18603606156815235584,833860238440653331505,39643749441387211150336

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

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

%t a[0] = 1; a[n_] := Sum[(n - 2*k)^(n - k), {k, 0, Floor[n/2]}]; Array[a, 20, 0] (* _Amiram Eldar_, Apr 16 2022 *)

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

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

%Y Cf. A031971, A353014.

%Y Cf. A352981, A352944.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Apr 16 2022