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

0,6

Links

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

Formula

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

Example

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, ...

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 *)

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.

Sequence in context: A246661 A246660 A346422 * A233543 A156588 A278543

Adjacent sequences: A245402 A245403 A245404 * A245406 A245407 A245408

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 21 2014

Status

approved