A245405 Number A(n,k) of endofunctions on [n] such that no element has a preimage of cardinality k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 0, 2, 1, 1, 2, 6, 1, 1, 2, 3, 24, 1, 1, 4, 9, 40, 120, 1, 1, 4, 24, 76, 205, 720, 1, 1, 4, 27, 208, 825, 2556, 5040, 1, 1, 4, 27, 252, 2325, 10206, 24409, 40320, 1, 1, 4, 27, 256, 3025, 31956, 143521, 347712, 362880, 1, 1, 4, 27, 256, 3120, 44406, 520723, 2313200, 4794633, 3628800

Offset: 0

Author

Alois P. Heinz, Jul 21 2014

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
0 :   1,    1,     1,     1,     1,     1,     1, ...
1 :   1,    0,     1,     1,     1,     1,     1, ...
2 :   2,    2,     2,     4,     4,     4,     4, ...
3 :   6,    3,     9,    24,    27,    27,    27, ...
4 :  24,   40,    76,   208,   252,   256,   256, ...
5 : 120,  205,   825,  2325,  3025,  3120,  3125, ...
6 : 720, 2556, 10206, 31956, 44406, 46476, 46650, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Column k=0-10 give: A000142, A231797, A245406, A245407, A245408, A245409, A245410, A245411, A245412, A245413, A245414.

Main diagonal gives A061190.

A(n,n+1) gives A000312.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 and i=0, 1,
      `if`(i<1, 0, add(`if`(j=k, 0, b(n-j, i-1, k)*
       binomial(n, j)), j=0..n)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

nn = n; f[m_]:=Flatten[Table[m[[j, i - j + 1]], {i, 1, Length[m]}, {j, 1, i}]]; f[Transpose[Table[Prepend[Table[n! Coefficient[Series[(Exp[x] -x^k/k!)^n, {x, 0, nn}],x^n], {n, 1, 10}], 1], {k, 0, 10}]]] (* Geoffrey Critzer, Jan 31 2015 *)

Formula

A(n,k) = n! * [x^n] (exp(x)-x^k/k!)^n.