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

1,2

Links

Table of n, a(n) for n=1..20.

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

Example

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.

Mathematica

Table[n!n(n^2+n+2)/4, {n, 20}] (* James C. McMahon, Feb 02 2024 *)

Crossrefs

Cf. A006595, A284816.

Sequence in context: A242631 A001090 A243782 * A105219 A060071 A037205

Adjacent sequences: A369807 A369808 A369809 * A369811 A369812 A369813

Keyword

nonn

Author

Ivaylo Kortezov, Feb 02 2024

Status

approved