This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369810 #13 Mar 02 2024 13:50:43
%S A369810 1,8,63,528,4800,47520,511560,5967360,75116160,1016064000,14709340800,
%T A369810 227046758400,3723758438400,64686292070400,1186714488960000,
%U A369810 22931377717248000,465594843377664000,9910874496466944000,220725034691825664000,5133423237252710400000
%N 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.
%F A369810 a(n) = n!*n*(n^2+n+2)/4.
%F A369810 a(n) = n*A284816(n).
%F A369810 a(n) = n^2*A006595(n-1).
%F A369810 E.g.f.: x*(2 + x^2)/(2*(1 - x)^4). - _Stefano Spezia_, Feb 05 2024
%e A369810 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.
%t A369810 Table[n!n(n^2+n+2)/4,{n,20}] (* _James C. McMahon_, Feb 02 2024 *)
%Y A369810 Cf. A006595, A284816.
%K A369810 nonn
%O A369810 1,2
%A A369810 _Ivaylo Kortezov_, Feb 02 2024