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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.
3
1, 2, 6, 3, 18, 36, 4, 36, 144, 240, 5, 60, 360, 1200, 1800, 6, 90, 720, 3600, 10800, 15120, 7, 126, 1260, 8400, 37800, 105840, 141120, 8, 168, 2016, 16800, 100800, 423360, 1128960, 1451520
OFFSET
1,2
COMMENTS
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
FORMULA
T(n,k) = binomial(k,2)*n!/(n+1-k)!
T(n,k) = n*T(n-1,k-1) + (k-1)*n!/(n+1-k)!
EXAMPLE
n=1 => AA -> T(1,2) = 1.
n=2 => AA, BB -> T(2,2) = 2; AAB, ABA, BAA, BBA, BAB, ABB -> T(2,3) = 6.
Triangle starts:
1
2, 6
3, 18, 36
4, 36, 144, 240
5, 60, 360, 1200, 1800
6, 90, 720, 3600, 10800, 15120
7, 126, 1260, 8400, 37800, 105840, 141120
8, 168, 2016, 16800, 100800, 423360, 1128960, 1451520
9, 216, 3024, 30240, 226800, 1270080, 5080320, 13063680, 16329600
10, 270, 4320, 50400, 453600, 3175200, 16934400, 65318400, 163296000, 199584000
MATHEMATICA
Table[Binomial[k, 2] n!/(n + 1 - k)!, {n, 8}, {k, 2, n + 1}] // Flatten (* Michael De Vlieger, Feb 02 2017 *)
CROSSREFS
Columns of table:
T(n,2) = A000027(n)
T(n,3) = A028896(n)
Other sequences in table:
T(n,n+1) = A001286(n)
T(n,n) = A001804(n), n>=2
Sequence in context: A276817 A050125 A178667 * A206493 A359256 A304085
KEYWORD
nonn,tabl
AUTHOR
Jeremy Dover, Feb 01 2017
STATUS
approved