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%I #22 May 27 2016 18:14:59
%S 1,1,1,2,6,1,6,45,12,1,24,420,160,20,1,120,4800,2450,375,30,1,720,
%T 65520,43050,7560,756,42,1,5040,1045170,858480,167825,19208,1372,56,1,
%U 40320,19126800,19208000,4110120,516096,43008,2304,72,1
%N 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 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 Alois P. Heinz, <a href="/A019576/b019576.txt">Rows n = 1..141, flattened</a>
%F T(n,k) = A019575(n,k)/n.
%e : 1;
%e : 1, 1;
%e : 2, 6, 1;
%e : 6, 45, 12, 1;
%e : 24, 420, 160, 20, 1;
%e : 120, 4800, 2450, 375, 30, 1;
%e : 720, 65520, 43050, 7560, 756, 42, 1;
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-j, i-1, k)/j!, j=0..min(k, n))))
%p end:
%p T:= (n, k)-> (n-1)!* (b(n$2, k) -b(n$2, k-1)):
%p seq(seq(T(n,k), k=1..n), n=1..12); # _Alois P. Heinz_, Jul 29 2014
%t 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 Row sums give A000169.
%Y Cf. A019575.
%K nonn,tabl,easy,nice
%O 1,4
%A Lee Corbin (lcorbin(AT)tsoft.com)
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