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

Author

Vaclav Kotesovec, Jul 24 2014

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

Cf. A231797, A245493, A245405, A331726.

Programs

Mathematica

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}]}]

PARI

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

PARI

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

Formula

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