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A349454
Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
5
1, 0, 1, 1, 0, 1, 8, 3, 0, 1, 81, 32, 6, 0, 1, 1024, 405, 80, 10, 0, 1, 15625, 6144, 1215, 160, 15, 0, 1, 279936, 109375, 21504, 2835, 280, 21, 0, 1, 5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1, 134217728, 51883209, 10077696, 1312500, 129024, 10206, 672, 36, 0, 1
OFFSET
0,7
LINKS
FORMULA
T(n,k) = binomial(n,k) * (n-k-1)^(n-k).
From Mélika Tebni, Apr 02 2023: (Start)
E.g.f. of column k: -x / (LambertW(-x)*(1+LambertW(-x)))*x^k / k!.
Sum_{k=0..n} k^k*T(n,k) = A217701(n). (End)
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
1, 0, 1;
8, 3, 0, 1;
81, 32, 6, 0, 1;
1024, 405, 80, 10, 0, 1;
15625, 6144, 1215, 160, 15, 0, 1;
279936, 109375, 21504, 2835, 280, 21, 0, 1;
5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1;
...
MAPLE
T:= (n, k)-> binomial(n, k)*(n-k-1)^(n-k):
seq(seq(T(n, k), k=0..n), n=0..10);
CROSSREFS
Column k=0 gives A065440.
Row sums give A204042.
Main diagonal and first lower diagonal give A000012, A000004.
T(n+1,n-1) gives A000217.
T(n+3,n) gives A130809.
T(n+3,n-1) gives A102741 for n>=1.
Sequence in context: A107671 A271174 A216891 * A256783 A154538 A154166
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Dec 30 2021
STATUS
approved