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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.
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%I #21 Jan 10 2018 15:19:37

%S 1,10,87,772,7285,74046,812875,9626632,122643657,1675253170,

%T 24449818591,379984902540,6268557335677,109443030279142,

%U 2016658652491155,39119860206021136,797013832285599505,17017679492994949722,380045072079456330727

%N 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.

%C 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.

%H Jeremy M. Dover, <a href="https://arxiv.org/abs/1710.06049">Repetition in Colored Sequences of Balls</a>, arXiv:1710.06049 [math.CO], 2017.

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

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

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

%o (PARI) a(n) = n! * sum(k=3, n+2, binomial(k,3)/(n+2-k)!); \\ _Michel Marcus_, Sep 29 2017

%Y Row sums of triangle A292930.

%Y Cf. A281912, A292999.

%K nonn

%O 1,2

%A _Jeremy Dover_, Sep 27 2017