A229243 Number A(n,k) of set partitions of {1,...,k*n} into sets of size at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 10, 5, 1, 1, 1, 76, 166, 15, 1, 1, 1, 764, 12644, 3795, 52, 1, 1, 1, 9496, 1680592, 3305017, 112124, 203, 1, 1, 1, 140152, 341185496, 6631556521, 1245131903, 4163743, 877, 1

Offset

0,9

Links

Alois P. Heinz, Antidiagonals n = 0..30, flattened

Formula

A(n,k) = (n*k)! * [x^(n*k)] exp(Sum_{j=1..n} x^j/j!).

A(n,k) = A229223(n*k,n).

Example

Square array A(n,k) begins:
  1,  1,      1,          1,              1,                   1, ...
  1,  1,      1,          1,              1,                   1, ...
  1,  2,     10,         76,            764,                9496, ...
  1,  5,    166,      12644,        1680592,           341185496, ...
  1, 15,   3795,    3305017,     6631556521,      25120541332271, ...
  1, 52, 112124, 1245131903, 41916097982471, 3282701194678476257, ...

Maple

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
A:= (n, k)-> G(n*k, n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n-k, k]; For[j = k-1, j >= 1, j--, g = g*(n-j)/j + G[n-j, k] ]; g ] ]; A[n_, k_] := G[n*k, n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 23 2013, translated from Maple *)

Crossrefs

Columns k=0-3 give: A000012, A000110, A229228, A229413.

Rows n=0+1, 2-3 give: A000012, A066223, A229414.

Main diagonal gives: A229229.

Cf. A229223.

Sequence in context: A156188 A179930 A007737 * A105688 A066017 A236938

Adjacent sequences: A229240 A229241 A229242 * A229244 A229245 A229246

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 17 2013

Status

approved