A353009
a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(n-2*k).
Original entry on oeis.org
1, 1, 5, 28, 261, 3153, 46917, 826696, 16824133, 388247185, 10016824133, 285699917796, 8926117272389, 303160806510049, 11120932942830405, 438197051187369424, 18457865006652382021, 827678458937524133601, 39364865940303189957445
Offset: 0
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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 *)
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a(n) = sum(k=0, n\2, (n-2*k)^(n-2*k));
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k)/(1-x^2))
A353016
a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(2*k).
Original entry on oeis.org
1, 1, 1, 2, 5, 11, 33, 108, 357, 1405, 5713, 24670, 117413, 574007, 3004577, 16608120, 95057925, 576245913, 3622049809, 23693870554, 161816447365, 1140392550275, 8351286979745, 63206781102116, 493344133444389, 3980464191557205, 33029872125113937, 282290255465835382
Offset: 0
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a[0] = 1; a[n_] := Sum[(n-2*k)^(2*k), {k, 0, Floor[n/2]}]; Array[a, 30, 0] (* Amiram Eldar, Apr 16 2022 *)
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a(n) = sum(k=0, n\2, (n-2*k)^(2*k));
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my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k*x)^2)))
A356834
a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^n/(n - 2*k)!.
Original entry on oeis.org
1, 1, 4, 33, 448, 8105, 192576, 5946913, 226097152, 10389920913, 571788928000, 36818407010561, 2741300619657216, 234014330510734969, 22620660476040331264, 2457467449742570271105, 298061856229112792743936, 40058727579693211737837857
Offset: 0
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f:= proc(n) local k; n! * add((n-2*k)^n/(n-2*k)!,k=0..floor(n/2)) end proc:
map(f, [$0..20]); # Robert Israel, Sep 16 2022
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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 *)
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a(n) = n!*sum(k=0, n\2, (n-2*k)^n/(n-2*k)!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^k/(k!*(1-(k*x)^2)))))
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