A292998 Number of sequences of balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence.

1, 10, 87, 772, 7285, 74046, 812875, 9626632, 122643657, 1675253170, 24449818591, 379984902540, 6268557335677, 109443030279142, 2016658652491155, 39119860206021136, 797013832285599505, 17017679492994949722, 380045072079456330727

Offset

1,2

Comments

Note that any such sequence has at least 3 balls and at most n+2, and that three matching balls must all be the same color.

Links

Table of n, a(n) for n=1..19.

Jeremy M. Dover, Repetition in Colored Sequences of Balls, arXiv:1710.06049 [math.CO], 2017.

Formula

a(n) = n! * Sum_{k=3..n+2} binomial(k,3)/(n+2-k)!.

Example

For n=2 colors a, b, the a(n)=10 sequences of balls are: aaa, bbb, abbb, babb, bbab, bbba, baaa, abaa, aaba, aaab.

Mathematica

Table[n!*Sum[Binomial[k, 3]/(n + 2 - k)!, {k, 3, n + 2}], {n, 19}] (* Michael De Vlieger, Sep 28 2017 *)

Prog

(PARI) a(n) = n! * sum(k=3, n+2, binomial(k, 3)/(n+2-k)!); \\ Michel Marcus, Sep 29 2017

Crossrefs

Row sums of triangle A292930.

Cf. A281912, A292999.

Sequence in context: A198858 A331793 A121115 * A114648 A350134 A217417

Adjacent sequences: A292995 A292996 A292997 * A292999 A293000 A293001

Keyword

nonn

Author

Jeremy Dover, Sep 27 2017

Status

approved