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A243098
Number T(n,k) of endofunctions on [n] with all cycles of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
8
1, 0, 1, 0, 3, 1, 0, 16, 6, 2, 0, 125, 51, 24, 6, 0, 1296, 560, 300, 120, 24, 0, 16807, 7575, 4360, 2160, 720, 120, 0, 262144, 122052, 73710, 41160, 17640, 5040, 720, 0, 4782969, 2285353, 1430016, 861420, 430080, 161280, 40320, 5040
OFFSET
0,5
COMMENTS
T(0,0) = 1 by convention.
LINKS
FORMULA
E.g.f. of column k>0: exp((-LambertW(-x))^k/k), e.g.f. of column k=0: 1.
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 16, 6, 2;
0, 125, 51, 24, 6;
0, 1296, 560, 300, 120, 24;
0, 16807, 7575, 4360, 2160, 720, 120;
0, 262144, 122052, 73710, 41160, 17640, 5040, 720;
...
MAPLE
with(combinat):
T:= (n, k)-> `if`(k*n=0, `if`(k+n=0, 1, 0),
add(binomial(n-1, j*k-1)*n^(n-j*k)*(k-1)!^j*
multinomial(j*k, k$j, 0)/j!, j=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
multinomial[n_, k_] := n!/Times @@ (k!); T[n_, k_] := If[k*n==0, If[k+n == 0, 1, 0], Sum[Binomial[n-1, j*k-1]*n^(n-j*k)*(k-1)!^j*multinomial[j*k, Append[Array[k&, j], 0]]/j!, {j, 0, n/k}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)
CROSSREFS
Columns k=0-4 give: A000007, A000272(n+1) for n>0, A057817(n+1), 2*A060917, 6*A060918.
Row sums give A241980.
T(2n,n) gives A246050.
Main diagonal gives A000142(n-1) for n>0.
Sequence in context: A131222 A228334 A114151 * A360177 A241981 A147723
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 18 2014
STATUS
approved