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
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
Programs
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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.
Comments