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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.

%I A229243 #21 Oct 07 2018 18:39:17
%S A229243 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 A229243 1,1,1,9496,1680592,3305017,112124,203,1,1,1,140152,341185496,
%U A229243 6631556521,1245131903,4163743,877,1
%N 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.
%H A229243 Alois P. Heinz, <a href="/A229243/b229243.txt">Antidiagonals n = 0..30, flattened</a>
%F A229243 A(n,k) = (n*k)! * [x^(n*k)] exp(Sum_{j=1..n} x^j/j!).
%F A229243 A(n,k) = A229223(n*k,n).
%e A229243 Square array A(n,k) begins:
%e A229243   1,  1,      1,          1,              1,                   1, ...
%e A229243   1,  1,      1,          1,              1,                   1, ...
%e A229243   1,  2,     10,         76,            764,                9496, ...
%e A229243   1,  5,    166,      12644,        1680592,           341185496, ...
%e A229243   1, 15,   3795,    3305017,     6631556521,      25120541332271, ...
%e A229243   1, 52, 112124, 1245131903, 41916097982471, 3282701194678476257, ...
%p A229243 G:= proc(n, k) option remember; local j; if k>n then G(n, n)
%p A229243       elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
%p A229243       for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
%p A229243     end:
%p A229243 A:= (n, k)-> G(n*k, n):
%p A229243 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A229243 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 A229243 Columns k=0-3 give: A000012, A000110, A229228, A229413.
%Y A229243 Rows n=0+1, 2-3 give: A000012, A066223, A229414.
%Y A229243 Main diagonal gives: A229229.
%Y A229243 Cf. A229223.
%K A229243 nonn,tabl
%O A229243 0,9
%A A229243 _Alois P. Heinz_, Sep 17 2013