A281881 -id:A281881 - cp's OEIS frontend

A281912 Number of sequences of balls colored with at most n colors such that exactly one ball is of a color seen earlier in the sequence.

1, 8, 57, 424, 3425, 30336, 294553, 3123632, 36003969, 448816600, 6022033721, 86587079448, 1328753602657, 21683227579664, 375013198304025, 6853321766162656, 131976208783240193, 2671430511854158632, 56709161712552286009, 1259836187316759240200

Offset: 1

Jeremy Dover, Feb 01 2017

nonn

Note that any such sequence has at least 2 balls, and at most n+1

Number of 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

			n=1 => AA -> a(1) = 1.
n=2 => AA,BB,AAB,ABA,BAA,BBA,BAB,ABB -> a(2) = 8.

Jeremy Dover, Table of n, a(n) for n = 1..99

Cf. A093964.

Row sums of triangle A281881. - Jeremy Dover, Sep 26 2017

a:= proc(n) option remember;
      `if`(n<2, 1, a(n-1)*(n+2)/(n-1)-a(n-2))*n
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Feb 02 2017

Table[n!*Sum[Binomial[k, 2]/(n + 1 - k)!, {k, 2, n + 1}], {n, 20}] (* Michael De Vlieger, Feb 02 2017 *)

a(n) = n! * Sum_{k=2..n+1} binomial(k,2)/(n+1-k)!.
a(n) = n if n < 2, a(n) = n*((n+2)/(n-1)*a(n-1) - a(n-2)) for n >= 2. - Alois P. Heinz, Feb 02 2017
a(n)/n! ~ e*n^2/2. - Vaclav Kotesovec, Feb 03 2017

A281944 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with n colors such that exactly two balls are of a color seen previously in the sequence.

Original entry on oeis.org

1, 2, 14, 3, 42, 150, 4, 84, 600, 1560, 5, 140, 1500, 7800, 16800, 6, 210, 3000, 23400, 100800, 191520, 7, 294, 5250, 54600, 352800, 1340640, 2328480, 8, 392, 8400, 109200, 940800, 5362560, 18627840, 30240000, 9, 504, 12600, 196560, 2116800, 16087680, 83825280, 272160000, 419126400, 10, 630, 18000, 327600, 4233600, 40219200, 279417600, 1360800000, 4191264000, 6187104000

Offset: 1

Jeremy Dover, Feb 02 2017

			n=1 => AAA -> T(1,3)=1
n=2 => AAA,BBB -> T(2,3)=2
   AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB,AABB,ABAB,ABBA,BAAB,BABA,BBAA -> T(2,4)=14
Triangle starts:
   1
   2,  14
   3,  42,   150
   4,  84,   600,   1560
   5, 140,  1500,   7800,   16800
   6, 210,  3000,  23400,  100800,   191520
   7, 294,  5250,  54600,  352800,  1340640,  2328480
   8, 392,  8400, 109200,  940800,  5362560, 18627840,  30240000
   9, 504, 12600, 196560, 2116800, 16087680, 83825280, 272160000, 419126400

Jeremy Dover, Table of n, a(n) for n = 1..1035
Jeremy Dover, Answer to Cumulative Distribution Function of Collision Counts

Columns of table: T(n,3) = A000027(n), T(n,4) = A163756(n).

Other sequences in table: T(n,n+2) = A037960(n).

Table[(Binomial[k, 3] + 3 Binomial[k, 4]) n!/(n + 2 - k)!, {n, 12}, {k, 3, n + 2}] // Flatten (* Michael De Vlieger, Feb 05 2017 *)

T(n, k) = (binomial(k,3) + 3*binomial(k,4)) * n! / (n+2-k)!;
tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Feb 04 2017

T(n, k) = (binomial(k,3) + 3*binomial(k,4)) * n! / (n+2-k)!.

T(n, k) = n*T(n-1,k-1) + (k-2)*A281881(n,k-1).

cp's OEIS Frontend

A281912 Number of sequences of balls colored with at most n colors such that exactly one ball is of a color seen earlier in the sequence.

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