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A019576 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives triangle of numbers f(n,k)/n.

%I A019576 #22 May 27 2016 18:14:59
%S A019576 1,1,1,2,6,1,6,45,12,1,24,420,160,20,1,120,4800,2450,375,30,1,720,
%T A019576 65520,43050,7560,756,42,1,5040,1045170,858480,167825,19208,1372,56,1,
%U A019576 40320,19126800,19208000,4110120,516096,43008,2304,72,1
%N A019576 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives triangle of numbers f(n,k)/n.
%C A019576 T(n,k) is 1/n times the number of endofunctions on [n] such that the maximal cardinality of the nonempty preimages equals k. - _Alois P. Heinz_, May 23 2016
%H A019576 Alois P. Heinz, <a href="/A019576/b019576.txt">Rows n = 1..141, flattened</a>
%F A019576 T(n,k) = A019575(n,k)/n.
%e A019576 :   1;
%e A019576 :   1,     1;
%e A019576 :   2,     6,     1;
%e A019576 :   6,    45,    12,    1;
%e A019576 :  24,   420,   160,   20,   1;
%e A019576 : 120,  4800,  2450,  375,  30,  1;
%e A019576 : 720, 65520, 43050, 7560, 756, 42, 1;
%p A019576 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A019576       add(b(n-j, i-1, k)/j!, j=0..min(k, n))))
%p A019576     end:
%p A019576 T:= (n, k)-> (n-1)!* (b(n$2, k) -b(n$2, k-1)):
%p A019576 seq(seq(T(n,k), k=1..n), n=1..12);  # _Alois P. Heinz_, Jul 29 2014
%t A019576 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1, k]/j!, {j, 0, Min[k, n]}]]]; T[n_, k_] := (n-1)!*(b[n, n, k]-b[n, n, k-1]); Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Jan 15 2015, after _Alois P. Heinz_ *)
%Y A019576 Row sums give A000169.
%Y A019576 Cf. A019575.
%K A019576 nonn,tabl,easy,nice
%O A019576 1,4
%A A019576 Lee Corbin (lcorbin(AT)tsoft.com)