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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.
6

%I #21 Oct 07 2018 18:39:17

%S 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,

%T 1,1,1,9496,1680592,3305017,112124,203,1,1,1,140152,341185496,

%U 6631556521,1245131903,4163743,877,1

%N 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.

%H Alois P. Heinz, <a href="/A229243/b229243.txt">Antidiagonals n = 0..30, flattened</a>

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

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

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 10, 76, 764, 9496, ...

%e 1, 5, 166, 12644, 1680592, 341185496, ...

%e 1, 15, 3795, 3305017, 6631556521, 25120541332271, ...

%e 1, 52, 112124, 1245131903, 41916097982471, 3282701194678476257, ...

%p G:= proc(n, k) option remember; local j; if k>n then G(n, n)

%p elif n=0 then 1 elif k<1 then 0 else G(n-k, k);

%p for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi

%p end:

%p A:= (n, k)-> G(n*k, n):

%p seq(seq(A(n, d-n), n=0..d), d=0..10);

%t 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 *)

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

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

%Y Main diagonal gives: A229229.

%Y Cf. A229223.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Sep 17 2013