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A245496
a(n) = n! * [x^n] (exp(x)+x)^n.
4
1, 2, 10, 87, 1096, 18045, 365796, 8793337, 244327616, 7701562377, 271493172100, 10582453248741, 451909972458000, 20980984760560045, 1052197311966267572, 56683993296812515425, 3264626390205804733696, 200168726219982496336401, 13017989155680578824221060
OFFSET
0,2
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
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
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}]}]
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Jul 24 2014
STATUS
approved