A356672
a(n) = n! * Sum_{k=0..n} k^(2*(n-k))/k!.
Original entry on oeis.org
1, 1, 3, 19, 253, 5661, 188191, 8983423, 594848409, 52174034713, 5852229698971, 822684190381131, 142739480367287893, 30074750245383836149, 7575373641076070706423, 2252600759590927171373431, 783103569459739402827046321, 315587346190678252431713684913
Offset: 0
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a(n) = n!*sum(k=0, n, k^(2*(n-k))/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k^2*x)))))
A356674
a(n) = n! * Sum_{k=0..n} k^(k*(n-k))/k!.
Original entry on oeis.org
1, 2, 5, 25, 349, 19941, 4440391, 4382699203, 17687865017481, 356274213630958297, 33338407933090938442411, 16214021627369697901867402911, 43817834057167927861655409052462093, 595284492835035398061242850538179192931525
Offset: 0
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Table[n!*(1 + Sum[k^(k*(n-k))/k!, {k, 1, n}]), {n, 0, 12}] (* Vaclav Kotesovec, Nov 27 2022 *)
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a(n) = n!*sum(k=0, n, k^(k*(n-k))/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k^k*x)))))
A354437
a(n) = n! * Sum_{k=0..n} (-k)^(n-k)/k!.
Original entry on oeis.org
1, 1, -1, 1, 13, -199, 2251, -19991, 7001, 7530193, -330734249, 11005284401, -300961551131, 4886902605001, 184195977487523, -28517140157423399, 2322376314679777201, -153646291657993064671, 8388000381774954552751, -287686436757241322569247
Offset: 0
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Join[{1}, Table[n!*Sum[ (-k)^(n - k)/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 28 2022 *)
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a(n) = n!*sum(k=0, n, (-k)^(n-k)/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1+k*x)))))
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from math import factorial
def A354437(n): return sum(factorial(n)*(-k)**(n-k)//factorial(k) for k in range(n+1)) # Chai Wah Wu, May 28 2022
A357146
a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^(2*k)/(n - 2*k)!.
Original entry on oeis.org
1, 1, 1, 7, 49, 301, 6241, 74131, 1722337, 46346329, 1090339201, 48905462431, 1584330498961, 81705172522117, 4191355357015009, 223743062044497451, 16563314120270608321, 1027165911865738200241, 91346158358120706564097, 7395168869747626389974839
Offset: 0
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a(n) = n!*sum(k=0, n\2, (n-2*k)^(2*k)/(n-2*k)!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-(k*x)^2)))))
A357147
a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^(3*k)/(n - 3*k)!.
Original entry on oeis.org
1, 1, 1, 1, 25, 481, 3241, 18481, 1332241, 44198785, 623190961, 15416707681, 1602405014761, 68167258954081, 1598025440555545, 134130467333575441, 14793638741719612321, 730659540435131811841, 34674365632872552887521, 5776415685538277157146305
Offset: 0
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a(n) = n!*sum(k=0, n\3, (n-3*k)^(3*k)/(n-3*k)!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-(k*x)^3)))))