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A232055
Triangular array read by rows: T(n,k) is the number of forests of rooted labeled trees such that the vertex labeled with 1 is in a component (rooted tree) of size k, n>=1, 1<=k<=n.
0
1, 1, 2, 3, 4, 9, 16, 18, 27, 64, 125, 128, 162, 256, 625, 1296, 1250, 1440, 1920, 3125, 7776, 16807, 15552, 16875, 20480, 28125, 46656, 117649, 262144, 235298, 244944, 280000, 350000, 489888, 823543, 2097152
OFFSET
1,3
COMMENTS
Column 1 is A000272.
T(n,n) = A000169(n).
T(n+1,n) = A000312(n).
T(n+2,n)/3 = A081132(n-1).
REFERENCES
Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill, 2007, page 282.
FORMULA
T(n,k) = binomial(n-1,k-1)*k^(k-1)*(n-k+1)^(n-k-1).
EXAMPLE
1;
1, 2;
3, 4, 9;
16, 18, 27, 64;
125, 128, 162, 256, 625;
1296, 1250, 1440, 1920, 3125, 7776;
MATHEMATICA
Table[Table[Binomial[n, k](k+1)(k+1)^(k-1)(n-k+1)^(n-k-1), {k, 0, n}], {n, 0, 7}]//Grid
CROSSREFS
Sequence in context: A049909 A346211 A096781 * A192818 A119721 A098969
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Nov 17 2013
STATUS
approved