A369810 Number of ways to color n+1 identical balls using n distinct colors (each color is used) and place them in n numbered cells so that each cell contains at least one ball.
1, 8, 63, 528, 4800, 47520, 511560, 5967360, 75116160, 1016064000, 14709340800, 227046758400, 3723758438400, 64686292070400, 1186714488960000, 22931377717248000, 465594843377664000, 9910874496466944000, 220725034691825664000, 5133423237252710400000
Offset: 1
Keywords
Examples
For n=3 one of the colors c (3 choices) is used twice and one of the cells k (3 choices) gets two balls. If the cell k does not contain a c-colored ball, then all other cells do (1 variant). If the cell k contains a c-colored ball, after its removal there are 3!=6 variants for placing the remaining 3 different balls in the 3 cells. In total there are 3*3*(1+6)=63 variants.
Programs
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Mathematica
Table[n!n(n^2+n+2)/4,{n,20}] (* James C. McMahon, Feb 02 2024 *)
Formula
a(n) = n!*n*(n^2+n+2)/4.
a(n) = n*A284816(n).
a(n) = n^2*A006595(n-1).
E.g.f.: x*(2 + x^2)/(2*(1 - x)^4). - Stefano Spezia, Feb 05 2024