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)))))
A357191
a(n) = n! * Sum_{k=0..floor(n/2)} k^n/k!.
Original entry on oeis.org
1, 0, 2, 6, 216, 2040, 111240, 2164680, 159391680, 5247305280, 491431600800, 24437592194400, 2800955712804480, 195393943295591040, 26699221909806526080, 2479967110139382864000, 396503602252401676032000, 47167550656581451928832000
Offset: 0
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a(n) = n!*sum(k=0, n\2, k^n/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^(2*k)/(k!*(1-k*x)))))
A357192
a(n) = n! * Sum_{k=0..floor(n/3)} k^n/k!.
Original entry on oeis.org
1, 0, 0, 6, 24, 120, 23760, 327600, 5201280, 1283688000, 37574409600, 1219438281600, 378254710310400, 19092171351052800, 1045282110435763200, 394211859168070944000, 30499777423295212032000, 2523689643597315088896000, 1125362204955051396299366400
Offset: 0
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a(n) = n!*sum(k=0, n\3, k^n/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^(3*k)/(k!*(1-k*x)))))
A357193
a(n) = n! * Sum_{k=0..floor(n/2)} k^(2*n)/k!.
Original entry on oeis.org
1, 0, 2, 6, 3096, 61560, 65248200, 4058986680, 7506140268480, 1062517243193280, 3052268000677879200, 822543740977513816800, 3395913346775619237617280, 1553795963458841732838848640, 8727392877498334693600263757440
Offset: 0
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a(n) = n!*sum(k=0, n\2, k^(2*n)/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*x)^(2*k)/(k!*(1-k^2*x)))))
A357194
a(n) = n! * Sum_{k=0..floor(n/3)} k^(3*n)/k!.
Original entry on oeis.org
1, 0, 0, 6, 24, 120, 94372560, 5284828080, 338228714880, 461220488356944960, 124524904888012809600, 36983489578531184304000, 94262861823240343196388902400, 78420937396722501660156363686400, 70262981254649802508019882162611200
Offset: 0
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a(n) = n!*sum(k=0, n\3, k^(3*n)/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^3*x)^(3*k)/(k!*(1-k^3*x)))))
A357174
a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^n/(n - 3*k)!.
Original entry on oeis.org
1, 1, 4, 27, 280, 5045, 134136, 4269223, 153188176, 6657007113, 371930499280, 25072409219891, 1872319689314856, 154583203638018493, 14784597239881491400, 1641532369038107170815, 201617558936011146124576, 26755058016106471234608017
Offset: 0
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a[n_] := n! * Sum[(n - 3*k)^n/(n - 3*k)!, {k, 0, Floor[n/3]} ]; a[0] = 1; Array[a, 18, 0] (* Amiram Eldar, Sep 16 2022 *)
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a(n) = n!*sum(k=0, n\3, (n-3*k)^n/(n-3*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)^3)))))
A368725
a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^n / k!.
Original entry on oeis.org
1, 1, 2, 9, 100, 1065, 10626, 224161, 4598504, 46288017, 2509940710, 84061763841, -1602021820596, 164372205860473, 5216105126641514, -883395389739028095, 79008645559978113616, -1023235751229436800735, -651030746777115881959602, 113943411938145511923004513
Offset: 0
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Join[{1}, Table[n!*Sum[(-1)^(n-k)*k^n/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jul 18 2025 *)
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a(n) = n!*sum(k=0, n, (-1)^(n-k)*k^n/k!);