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A292999 Triangle read by rows: T(n,k) (n >= 1, 4 <= k <= n+3) is the number of k-sequences of balls colored with at most n colors such that exactly four balls are the same color as some other ball in the sequence. 1
1, 8, 10, 21, 120, 90, 40, 420, 1440, 840, 65, 1000, 6300, 16800, 8400, 96, 1950, 18000, 88200, 201600, 90720, 133, 3360, 40950, 294000, 1234800, 2540160, 1058400, 176, 5320, 80640, 764400, 4704000, 17781120, 33868800, 13305600, 225, 7920, 143640, 1693440, 13759200, 76204800, 266716800, 479001600, 179625600 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..45.

FORMULA

a(n) = binomial(k,4)*n!*(1/(n+3-k)! + 3/(n+2-k)!) (with the convention that 3/(-1)! = 0 when k=n+3).

EXAMPLE

For n=1: AAAA -> T(1,4)=1.

For n=2: AAAA,BBBB,AABB,ABAB,ABBA,BAAB,BABA,BBAA -> T(2,4)=8; AAAAB,AAABA,AABAA,ABAAA,BAAAA,BBBBA,BBBAB,BBABB,BABBB,ABBBB -> T(2,5)=10.

Triangle starts:

    1;

    8,   10;

   21,  120,     90;

   40,  420,   1440,     840;

   65, 1000,   6300,   16800,     8400;

   96, 1950,  18000,   88200,   201600,    90720;

  133, 3360,  40950,  294000,  1234800,  2540160,   1058400;

  176, 5320,  80640,  764400,  4704000, 17781120,  33868800, 13305600;

  225, 7920, 143640, 1693440, 13759200, 76204800, 266716800, ... .

MATHEMATICA

Table[Binomial[k, 4] n! (1/(n + 3 - k)! + 3/(n + 2 - k)!), {n, 9}, {k, 4, n + 3}] // Flatten (* Michael De Vlieger, Sep 30 2017 *)

CROSSREFS

Columns of the table: T(n,4) = A000567(n), T(n,5) = 10*A007586(n-1), T(n,6) = 90*A220212(n-2).

Diagonals of the table: T(n,n+3) = A061206(n), T(n+1,n+3) = 8*A005461(n), T(n-1,n) = 21*A001755(n), T(n,n) = 40*A001811(n), T(n,n-1) = 65*A001777(n), T(n+6,n+4) = A062194(n).

Sequence in context: A073619 A338820 A302429 * A216047 A032488 A102844

Adjacent sequences:  A292996 A292997 A292998 * A293000 A293001 A293002

KEYWORD

nonn,tabl

AUTHOR

Jeremy Dover, Sep 27 2017

STATUS

approved

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Last modified November 24 07:53 EST 2020. Contains 338607 sequences. (Running on oeis4.)