cp's OEIS Frontend

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.

%I A243098 #28 Dec 16 2021 16:47:48
%S A243098 1,0,1,0,3,1,0,16,6,2,0,125,51,24,6,0,1296,560,300,120,24,0,16807,
%T A243098 7575,4360,2160,720,120,0,262144,122052,73710,41160,17640,5040,720,0,
%U A243098 4782969,2285353,1430016,861420,430080,161280,40320,5040
%N 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.
%C A243098 T(0,0) = 1 by convention.
%H A243098 Alois P. Heinz, <a href="/A243098/b243098.txt">Rows n = 0..140, flattened</a>
%F A243098 E.g.f. of column k>0: exp((-LambertW(-x))^k/k), e.g.f. of column k=0: 1.
%e A243098 Triangle T(n,k) begins:
%e A243098   1;
%e A243098   0,      1;
%e A243098   0,      3,      1;
%e A243098   0,     16,      6,     2;
%e A243098   0,    125,     51,    24,     6;
%e A243098   0,   1296,    560,   300,   120,    24;
%e A243098   0,  16807,   7575,  4360,  2160,   720,  120;
%e A243098   0, 262144, 122052, 73710, 41160, 17640, 5040, 720;
%e A243098   ...
%p A243098 with(combinat):
%p A243098 T:= (n, k)-> `if`(k*n=0, `if`(k+n=0, 1, 0),
%p A243098     add(binomial(n-1, j*k-1)*n^(n-j*k)*(k-1)!^j*
%p A243098     multinomial(j*k, k$j, 0)/j!, j=0..n/k)):
%p A243098 seq(seq(T(n, k), k=0..n), n=0..10);
%t A243098 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 *)
%Y A243098 Columns k=0-4 give: A000007, A000272(n+1) for n>0, A057817(n+1), 2*A060917, 6*A060918.
%Y A243098 Row sums give A241980.
%Y A243098 T(2n,n) gives A246050.
%Y A243098 Main diagonal gives A000142(n-1) for n>0.
%Y A243098 Cf. A241981, A246049.
%K A243098 nonn,tabl
%O A243098 0,5
%A A243098 _Alois P. Heinz_, Aug 18 2014