

A292930


Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of ksequences 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



1, 2, 8, 3, 24, 60, 4, 48, 240, 480, 5, 80, 600, 2400, 4200, 6, 120, 1200, 7200, 25200, 40320, 7, 168, 2100, 16800, 88200, 282240, 423360, 8, 224, 3360, 33600, 235200, 1128960, 3386880, 4838400, 9, 288, 5040, 60480, 529200, 3386880, 15240960, 43545600, 59875200, 10, 360, 7200, 100800, 1058400, 8467200, 50803200, 217728000, 598752000, 798336000
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OFFSET

1,2


COMMENTS

Note that the three matching balls are necessarily the same color.


LINKS

Table of n, a(n) for n=1..55.


FORMULA

T(n, k) = binomial(k,3)*n!/(n+2k)!.


EXAMPLE

n=1 => AAA > T(1,3)=1;
n=2 => AAA,BBB > T(2,3)=2;
AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB > T(2,4)=8.
Triangle begins:
1;
2, 8;
3, 24, 60;
4, 48, 240, 480;
5, 80, 600, 2400, 4200;
...


PROG

(PARI) T(n, k) = binomial(k, 3)*n!/(n+2k)!;
tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n, k), ", ")); print()); \\ Michel Marcus, Sep 29 2017


CROSSREFS

Columns of table: T(n,3) = A000027(n), T(n,4) = A033996(n).
Other sequences in table: T(n,n+2) = A005990(n+1).
Sequence in context: A153188 A193976 A264244 * A126951 A282637 A256411
Adjacent sequences: A292927 A292928 A292929 * A292931 A292932 A292933


KEYWORD

nonn,tabl


AUTHOR

Jeremy Dover, Sep 26 2017


STATUS

approved



