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a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(n-2*k).
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%I #57 Apr 17 2022 01:13:58

%S 1,1,5,28,261,3153,46917,826696,16824133,388247185,10016824133,

%T 285699917796,8926117272389,303160806510049,11120932942830405,

%U 438197051187369424,18457865006652382021,827678458937524133601,39364865940303189957445

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

%H Seiichi Manyama, <a href="/A353009/b353009.txt">Table of n, a(n) for n = 0..386</a>

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

%F a(2*n-1) = A061787(n), a(2*n) = A061788(n) + 1. - _Seiichi Manyama_, Apr 17 2022

%t a[n_] := Sum[If[2*k == n, 1, (n - 2*k)^(n - 2*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-2*k));

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

%Y Cf. A062970, A353018.

%Y Cf. A061787, A061788, A352082, A353013.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Apr 16 2022