A245496 - OEIS

%I #23 Jul 19 2022 11:53:01

%S 1,2,10,87,1096,18045,365796,8793337,244327616,7701562377,

%T 271493172100,10582453248741,451909972458000,20980984760560045,

%U 1052197311966267572,56683993296812515425,3264626390205804733696,200168726219982496336401,13017989155680578824221060

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

%C 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

%H Vincenzo Librandi, <a href="/A245496/b245496.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ (1+exp(-1))^(n+1/2) * n^n.

%F E.g.f.: 1 / ((1 - x) * (1 + LambertW(-x/(1 - x)))). - _Ilya Gutkovskiy_, Jan 25 2020

%F a(n) = n! * Sum_{k=0..n} k^k/k! * binomial(n,k). - _Seiichi Manyama_, Jul 19 2022

%t Table[n!*SeriesCoefficient[(E^x+x)^n, {x, 0, n}], {n, 0, 20}]

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

%o (PARI) seq(n)={Vec(serlaplace(1/((1 - x) * (1 + lambertw(-x/(1 - x) + O(x*x^n))))), -(n+1))} \\ _Andrew Howroyd_, Jan 25 2020

%o (PARI) a(n) = n!*sum(k=0, n, k^k/k!*binomial(n, k)); \\ _Seiichi Manyama_, Jul 19 2022

%Y Cf. A231797, A245493, A245405, A331726.

%K nonn,easy

%O 0,2

%A _Vaclav Kotesovec_, Jul 24 2014