A245493 -id:A245493 - cp's OEIS frontend

A245496 a(n) = n! * [x^n] (exp(x)+x)^n.

1, 2, 10, 87, 1096, 18045, 365796, 8793337, 244327616, 7701562377, 271493172100, 10582453248741, 451909972458000, 20980984760560045, 1052197311966267572, 56683993296812515425, 3264626390205804733696, 200168726219982496336401, 13017989155680578824221060

Offset: 0

Vaclav Kotesovec, Jul 24 2014

a(n) is the number of ways to place n labeled balls (colored red and blue) into n labeled bins so that if a blue ball occupies a bin then there are no other balls with it. - Geoffrey Critzer, Jan 30 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A231797, A245493, A245405, A331726.

Table[n!*SeriesCoefficient[(E^x+x)^n, {x, 0, n}], {n, 0, 20}]
Flatten[{1,Table[n!+Sum[Binomial[n,j]^2*(n-j)^(n-j)*j!,{j,0,n-1}],{n,1,20}]}]

seq(n)={Vec(serlaplace(1/((1 - x) * (1 + lambertw(-x/(1 - x) + O(x*x^n))))), -(n+1))} \\ Andrew Howroyd, Jan 25 2020

a(n) = n!*sum(k=0, n, k^k/k!*binomial(n, k)); \\ Seiichi Manyama, Jul 19 2022

a(n) ~ (1+exp(-1))^(n+1/2) * n^n.
E.g.f.: 1 / ((1 - x) * (1 + LambertW(-x/(1 - x)))). - Ilya Gutkovskiy, Jan 25 2020
a(n) = n! * Sum_{k=0..n} k^k/k! * binomial(n,k). - Seiichi Manyama, Jul 19 2022

A245406 Number of endofunctions on [n] such that no element has a preimage of cardinality two.

Original entry on oeis.org

1, 1, 2, 9, 76, 825, 10206, 143521, 2313200, 42482313, 875799550, 19972186311, 498430219464, 13509979971241, 395352049852046, 12425644029361725, 417456939168255616, 14929305882415781265, 566234625018001351230, 22701936510037394021395, 959341639105178919209000

Offset: 0

Alois P. Heinz, Jul 21 2014

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..396 (first 200 terms from Alois P. Heinz)

Column k=2 of A245405.

Cf. A245493.

b:= proc(n, i) option remember; `if`(n=0 and i=0, 1, `if`(i<1, 0,
       add(`if`(j=2, 0, b(n-j, i-1) *binomial(n, j)), j=0..n)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

Table[n!*SeriesCoefficient[(E^x - x^2/2)^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 23 2014 *)
With[{k=2},Flatten[{1,Table[Sum[Binomial[n,j]*Binomial[n,k*j]*(-1)^j*(n-j)^(n-k*j)*(k*j)!/(k!)^j,{j,0,n/k}],{n,1,20}]}]]  (* Vaclav Kotesovec, Jul 24 2014 *)

a(n) = n! * [x^n] (exp(x)-x^2/2!)^n.

a(n) ~ c * d^n * n^n / exp(n), where d = 2.166383277092734585444028653747119..., c = 0.8627963719760750933657356839596... . - Vaclav Kotesovec, Jul 24 2014

cp's OEIS Frontend

A245496 a(n) = n! * [x^n] (exp(x)+x)^n.

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