A210457 Triangular array read by rows: T(n,k) is the number of elements x in {1,2,...,n} such that |(f^-1)(x)| = k over all functions f:{1,2,...,n}->{1,2,...,n}; n>=0, 0<=k<=n.

0, 0, 1, 2, 4, 2, 24, 36, 18, 3, 324, 432, 216, 48, 4, 5120, 6400, 3200, 800, 100, 5, 93750, 112500, 56250, 15000, 2250, 180, 6, 1959552, 2286144, 1143072, 317520, 52920, 5292, 294, 7, 46118408, 52706752, 26353376, 7529536, 1344560, 153664, 10976, 448, 8

Offset

0,4

Comments

Row sums = n^(n+1) = Sum_{k=1..n} T(n,k)*k.

Column for k=0 is A209290.

Distribution is Poisson with mean = 1.

Links

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

Formula

T(n,k) = binomial(n,k)*(n-1)^(n-k)*n.

Example

0;
0,       1;
2,       4,       2;
24,      36,      18,      3;
324,     432,     216,     48,     4;
5120,    6400,    3200,    800,    100,   5;
93750,   112500,  56250,   15000,  2250,  180,  6;
1959552, 2286144, 1143072, 317520, 52920, 5292, 294, 7;

Mathematica

nn=7; f[list_]:=Sum[list[[i]]*(i-1), {i, 1, Length[list]}]; g[list_]:= Select[list, #>0&]; Prepend[Insert[Map[g, Transpose[Table[Table[f[ CoefficientList[n!Coefficient[Series[(Exp[x]-x^m/m!+y x^m/m!)^n, {x, 0, nn}], x^n], y]], {n, 1, nn}], {m, 0, nn}]]], 0, {1, 1}], {0}]//Grid

Crossrefs

Sequence in context: A295390 A059890 A295640 * A006496 A352642 A361077

Adjacent sequences: A210454 A210455 A210456 * A210458 A210459 A210460

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jan 21 2013

Status

approved