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.

1, 0, 1, 1, 2, 1, 14, 12, 0, 1, 181, 68, 6, 0, 1, 2584, 520, 20, 0, 0, 1, 41973, 4542, 120, 20, 0, 0, 1, 776250, 46550, 672, 70, 0, 0, 0, 1, 16231381, 540136, 5516, 112, 70, 0, 0, 0, 1, 380333228, 7045020, 40140, 1848, 252, 0, 0, 0, 0, 1

Offset

0,5

Comments

T(0,0) = 1 by convention.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

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!).

T(n,k) = A245732(n,k) - A245732(n,k+1).

Example

T(2,0) = 1: (2,2).
T(2,1) = 2: (1,2), (2,1).
T(2,2) = 1: (1,1).
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).
T(3,3) = 1: (1,1,1).
Triangle T(n,k) begins:
0 :         1;
1 :         0,      1;
2 :         1,      2,    1;
3 :        14,     12,    0,   1;
4 :       181,     68,    6,   0,  1;
5 :      2584,    520,   20,   0,  0, 1;
6 :     41973,   4542,  120,  20,  0, 0, 1;
7 :    776250,  46550,  672,  70,  0, 0, 0, 1;
8 :  16231381, 540136, 5516, 112, 70, 0, 0, 0, 1;
     ...

Maple

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
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):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

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 *)

Crossrefs

Columns k=0-10 give: A133286 (for n>0), A245854, A245855, A245856, A245857, A245858, A245859, A245860, A245861, A245862, A245863.

Row sums give A000312.

T(2n,n) gives A000984(n).

Cf. A245732.

Sequence in context: A307804 A338207 A063613 * A276851 A080346 A216445

Adjacent sequences: A245730 A245731 A245732 * A245734 A245735 A245736

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 30 2014

Status

approved