A351768 -id:A351768 - cp's OEIS frontend

A351765 a(n) = n! * Sum_{k=0..n} n^(n-k) * (n-k)^k/k!.

1, 1, 12, 279, 11536, 746525, 69768036, 8902181575, 1487939919936, 315597946293657, 82839437215344100, 26366747854082944451, 10006618140321691249296, 4464690010732922712332149, 2313871692128866349730705924, 1378552938661073773617331110975

Offset: 0

Seiichi Manyama, Feb 18 2022

nonn

Main diagonal of A351761.

Cf. A332408, A351768, A351779.

Join[{1}, Table[n!*Sum[n^(n - k)*(n - k)^k/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Feb 19 2022 *)

a(n) = n!*sum(k=0, n, n^(n-k)*(n-k)^k/k!);

a(n) = n! * [x^n] 1/(1 - n*x*exp(x)).
From Vaclav Kotesovec, Feb 19 2022: (Start)
a(n) ~ exp(1) * n! * n^n.
a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n-1). (End)

A351780 a(n) = n! * Sum_{k=0..n} (-k)^(n-k) * (n-k)^k/k!.

Original entry on oeis.org

1, 0, -2, 6, 108, -2420, 8730, 1313718, -57930152, 567983736, 109544982390, -9917916180590, 321821829728388, 32383946348733252, -6591798188344856942, 531702135210365508270, 11136706526396459006640

Offset: 0

Seiichi Manyama, Feb 19 2022

sign

Cf. A351768, A351779.

a(n) = n!*sum(k=0, n, (-k)^(n-k)*(n-k)^k/k!);

A351795 a(n) = n! * Sum_{k=0..n} (k * (n-k))^k/k!.

Original entry on oeis.org

1, 1, 4, 30, 396, 8360, 256470, 10619952, 564959528, 37370475648, 3001942868490, 287388158562560, 32278318416029532, 4197544986996581376, 625014083479647028622, 105554855135062180485120, 20053957030647088382195280, 4255329207190209023134564352

Offset: 0

Seiichi Manyama, Feb 19 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..266
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A235328, A349965, A351768, A351796.

a[n_] := n!*(1 + Sum[(k*(n - k))^k/k!, {k, 1, n}]); Array[a, 18, 0] (* Amiram Eldar, Feb 19 2022 *)

PARI
```
a(n) = n!*sum(k=0, n, (k*(n-k))^k/k!);
```

a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / (sqrt(LambertW(exp(2)*n)^2 - 1) * exp(n*(1 - 1/LambertW(exp(2)*n))) * LambertW(exp(2)*n)^n). - Vaclav Kotesovec, Feb 20 2022

cp's OEIS Frontend

A351765 a(n) = n! * Sum_{k=0..n} n^(n-k) * (n-k)^k/k!.

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A351780 a(n) = n! * Sum_{k=0..n} (-k)^(n-k) * (n-k)^k/k!.

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A351795 a(n) = n! * Sum_{k=0..n} (k * (n-k))^k/k!.

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