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A245687 Number T(n,k) of endofunctions on [n] such that the minimal cardinality of the nonempty preimages equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A245687 #25 Dec 16 2021 16:49:29
%S A245687 1,0,1,0,2,2,0,24,0,3,0,216,36,0,4,0,2920,200,0,0,5,0,44100,2250,300,
%T A245687 0,0,6,0,799134,22932,1470,0,0,0,7,0,16429504,342608,3136,1960,0,0,0,
%U A245687 8,0,382625856,4638384,147168,9072,0,0,0,0,9,0,9918836100,79610850,1522800,18900,11340,0,0,0,0,10
%N A245687 Number T(n,k) of endofunctions on [n] such that the minimal cardinality of the nonempty preimages equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A245687 T(0,0) = 1 by convention.
%H A245687 Alois P. Heinz, <a href="/A245687/b245687.txt">Rows n = 0..100, flattened</a>
%e A245687 Triangle T(n,k) begins:
%e A245687   1;
%e A245687   0,        1;
%e A245687   0,        2,      2;
%e A245687   0,       24,      0,    3;
%e A245687   0,      216,     36,    0,    4;
%e A245687   0,     2920,    200,    0,    0,  5;
%e A245687   0,    44100,   2250,  300,    0,  0,  6;
%e A245687   0,   799134,  22932, 1470,    0,  0,  0,  7;
%e A245687   0, 16429504, 342608, 3136, 1960,  0,  0,  0,  8;
%e A245687   ...
%p A245687 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A245687       b(n, i-1, k) +add(b(n-j, i-1, k)/j!, j=k..n)))
%p A245687     end:
%p A245687 T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), `if`(k=n, n,
%p A245687     `if`(k>=(n+1)/2, 0, n!*(b(n$2, k)-b(n$2, k+1))))):
%p A245687 seq(seq(T(n, k), k=0..n), n=0..12);
%t A245687 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-j, i-1, k]/j!, {j, k, n}]]]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], If[k == n, n, If[k >= (n+1)/2, 0, n!*(b[n, n, k] - b[n, n, k+1])]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 02 2015, after _Alois P. Heinz_ *)
%Y A245687 T(n,1) = n*A241581(n) for n>0.
%Y A245687 Rows sums give A000312.
%Y A245687 Main diagonal gives A028310.
%Y A245687 T(2n,n) gives A273325.
%Y A245687 Cf. A019575 (the same for maximal cardinality).
%K A245687 nonn,tabl
%O A245687 0,5
%A A245687 _Alois P. Heinz_, Jul 29 2014