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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..30, flattened
Crossrefs
Programs
-
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 *)
Formula
A(n,k) = (n*k)! * [x^(n*k)] exp(Sum_{j=1..n} x^j/j!).
A(n,k) = A229223(n*k,n).