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A245733 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality k exists and, if j is the largest value with a nonempty preimage, the preimage cardinality of i is >=k for all i<=j and equal to k for at least one i<=j; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A245733 #29 Dec 16 2021 16:46:39
%S A245733 1,0,1,1,2,1,14,12,0,1,181,68,6,0,1,2584,520,20,0,0,1,41973,4542,120,
%T A245733 20,0,0,1,776250,46550,672,70,0,0,0,1,16231381,540136,5516,112,70,0,0,
%U A245733 0,1,380333228,7045020,40140,1848,252,0,0,0,0,1
%N A245733 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality k exists and, if j is the largest value with a nonempty preimage, the preimage cardinality of i is >=k for all i<=j and equal to k for at least one i<=j; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A245733 T(0,0) = 1 by convention.
%H A245733 Alois P. Heinz, <a href="/A245733/b245733.txt">Rows n = 0..140, flattened</a>
%F A245733 E.g.f. of column k=0: 1 +1/(1+LambertW(-x)) -1/(2-exp(x)); e.g.f. of column k>0: 1/(1-Sum_{j>=k} x^j/j!) - 1/(1-Sum_{j>=k+1} x^j/j!).
%F A245733 T(n,k) = A245732(n,k) - A245732(n,k+1).
%e A245733 T(2,0) = 1: (2,2).
%e A245733 T(2,1) = 2: (1,2), (2,1).
%e A245733 T(2,2) = 1: (1,1).
%e A245733 T(3,1) = 12: (1,1,2), (1,2,1), (1,2,2), (1,2,3), (1,3,2), (2,1,1), (2,1,2), (2,1,3), (2,2,1), (2,3,1), (3,1,2), (3,2,1).
%e A245733 T(3,3) = 1: (1,1,1).
%e A245733 Triangle T(n,k) begins:
%e A245733 0 :         1;
%e A245733 1 :         0,      1;
%e A245733 2 :         1,      2,    1;
%e A245733 3 :        14,     12,    0,   1;
%e A245733 4 :       181,     68,    6,   0,  1;
%e A245733 5 :      2584,    520,   20,   0,  0, 1;
%e A245733 6 :     41973,   4542,  120,  20,  0, 0, 1;
%e A245733 7 :    776250,  46550,  672,  70,  0, 0, 0, 1;
%e A245733 8 :  16231381, 540136, 5516, 112, 70, 0, 0, 0, 1;
%e A245733      ...
%p A245733 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A245733       add(b(n-j, k)*binomial(n, j), j=k..n))
%p A245733     end:
%p A245733 g:= (n, k)-> `if`(k=0, n^n, `if`(n=0, 0, b(n, k))):
%p A245733 T:= (n, k)-> g(n, k) -g(n, k+1):
%p A245733 seq(seq(T(n, k), k=0..n), n=0..12);
%t A245733 b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n-j, k]*Binomial[n, j], {j, k, n}]]; g[n_, k_] := If[k == 0, n^n, If[n == 0, 0, b[n, k]]]; T[n_, k_] := g[n, k] - g[n, k+1]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 27 2015, after _Alois P. Heinz_ *)
%Y A245733 Columns k=0-10 give: A133286 (for n>0), A245854, A245855, A245856, A245857, A245858, A245859, A245860, A245861, A245862, A245863.
%Y A245733 Row sums give A000312.
%Y A245733 T(2n,n) gives A000984(n).
%Y A245733 Cf. A245732.
%K A245733 nonn,tabl
%O A245733 0,5
%A A245733 _Alois P. Heinz_, Jul 30 2014