A331345 a(n) = (1/n^2) * Sum_{k>=1} k^n * (1 - 1/n)^(k - 1).

1, 3, 37, 1015, 48601, 3583811, 376372333, 53343571695, 9808511445361, 2270198126932219, 645790373135121061, 221449391959470686375, 90084675298978081317961, 42890688646618728144279987, 23627228721958495690763944861, 14910259060767841554203065990111

Offset: 1

Author

Ilya Gutkovskiy, Jun 08 2020

Keywords

• nonn

Links

• Index entries for sequences related to rooted trees

Crossrefs

Cf. A000670, A050351, A050352, A050353, A094420, A257565, A321189, A334240.

Programs

Mathematica

Join[{1}, Table[1/n^2 Sum[k^n (1 - 1/n)^(k - 1), {k, 1, Infinity}], {n, 2, 16}]]
Table[n! SeriesCoefficient[(Exp[x] - 1)/(Exp[x] - n (Exp[x] - 1)), {x, 0, n}], {n, 1, 16}]

Formula

a(n) = n! * [x^n] (exp(x) - 1) / (exp(x) - n * (exp(x) - 1)).

a(n) = Sum_{k=1..n} Stirling2(n,k) * (n - 1)^(k - 1) * k!.

a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jun 08 2020