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A281881 Triangle read by rows: T(n,k) (n>=1, 2<=k<=n+1) is the number of k-sequences of balls colored with at most n colors such that exactly one ball is of a color seen previously in the sequence.

%I A281881 #28 Nov 20 2024 04:18:43
%S A281881 1,2,6,3,18,36,4,36,144,240,5,60,360,1200,1800,6,90,720,3600,10800,
%T A281881 15120,7,126,1260,8400,37800,105840,141120,8,168,2016,16800,100800,
%U A281881 423360,1128960,1451520
%N A281881 Triangle read by rows: T(n,k) (n>=1, 2<=k<=n+1) is the number of k-sequences of balls colored with at most n colors such that exactly one ball is of a color seen previously in the sequence.
%C A281881 Number of k-sequences of balls colored with at most n colors such that exactly two balls are the same color as some other ball in the sequence (necessarily each other). - _Jeremy Dover_, Sep 26 2017
%H A281881 Jeremy Dover, <a href="/A281881/b281881.txt">Table of n, a(n) for n = 1..999</a>
%H A281881 Jeremy Dover, <a href="http://math.stackexchange.com/a/2109013/342250">Answer to Cumulative Distribution Function of Collision Counts</a>
%F A281881 T(n,k) = binomial(k,2)*n!/(n+1-k)!.
%F A281881 T(n,k) = n*T(n-1,k-1) + (k-1)*n!/(n+1-k)!.
%e A281881 n=1 => AA -> T(1,2) = 1.
%e A281881 n=2 => AA, BB -> T(2,2) = 2; AAB, ABA, BAA, BBA, BAB, ABB -> T(2,3) = 6.
%e A281881 Triangle starts:
%e A281881    1
%e A281881    2,   6
%e A281881    3,  18,   36
%e A281881    4,  36,  144,   240
%e A281881    5,  60,  360,  1200,   1800
%e A281881    6,  90,  720,  3600,  10800,   15120
%e A281881    7, 126, 1260,  8400,  37800,  105840,   141120
%e A281881    8, 168, 2016, 16800, 100800,  423360,  1128960,  1451520
%e A281881    9, 216, 3024, 30240, 226800, 1270080,  5080320, 13063680,  16329600
%e A281881   10, 270, 4320, 50400, 453600, 3175200, 16934400, 65318400, 163296000, 199584000
%t A281881 Table[Binomial[k, 2] n!/(n + 1 - k)!, {n, 8}, {k, 2, n + 1}] // Flatten (* _Michael De Vlieger_, Feb 02 2017 *)
%Y A281881 Columns of table:
%Y A281881   T(n,2) = A000027(n)
%Y A281881   T(n,3) = A028896(n)
%Y A281881 Other sequences in table:
%Y A281881   T(n,n+1) = A001286(n)
%Y A281881   T(n,n) = A001804(n), n>=2
%K A281881 nonn,tabl
%O A281881 1,2
%A A281881 _Jeremy Dover_, Feb 01 2017