|
%I #35 Jul 21 2021 15:01:38
%S 1,1,1,2,3,5,5,9,18,40,15,31,70,172,457,52,120,299,801,2295,6995,203,
%T 514,1393,4025,12347,40043,136771,877,2407,7023,21709,70843,243235,
%U 875936,3299218,4140,12205,38043,124997,431636,1562071,5908978,23308546,95668354
%N Array read by upwards antidiagonals: T(m,n) = number of set partitions into distinct parts of the multiset consisting of one copy each of x_1, x_2, ..., x_m, and two copies each of y_1, y_2, ..., y_n, for m >= 0, n >= 0.
%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. (Background information.)
%H Alois P. Heinz, <a href="/A322770/b322770.txt">Antidiagonals n = 0..140, flattened</a>
%H D. E. Knuth, <a href="/A322770/a322770_1.txt">Partitioning a multiset into submultisets</a>, Email to N. J. A. Sloane, Dec 29 2018.
%F Knuth gives a recurrence using the Bell numbers A000110 (see Maple program).
%e The array begins:
%e 1, 1, 5, 40, 457, 6995, 136771, ...
%e 1, 3, 18, 172, 2295, 40043, 875936, ...
%e 2, 9, 70, 801, 12347, 243235, 5908978, ...
%e 5, 31, 299, 4025, 70843, 1562071, 41862462, ...
%e 15, 120, 1393, 21709, 431636, 10569612, 310606617, ...
%e 52, 514, 7023, 124997, 2781372, 75114998, 2407527172, ...
%e 203, 2407, 38043, 764538, 18885177, 559057663, 19449364539, ...
%e ...
%p B := n -> combinat[bell](n):
%p Q := proc(m,n) local k; global B; option remember;
%p if n = 0 then B(m) else
%p (1/2)*( Q(m+2,n-1) + Q(m+1,n-1) - add( binomial(n-1,k)*Q(m,k), k=0..n-1) ); fi; end; # Q(m,n) (which is Knuth's notation) is T(m,n)
%t Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2)(Q[m+2, n-1] + Q[m+1, n-1] - Sum[Binomial[n-1, k] Q[m, k], {k, 0, n-1}])];
%t Table[Q[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* _Jean-François Alcover_, Jan 02 2019, from Maple *)
%Y Rows include A094574, A322771, A322772.
%Y Columns include A000110, A087648, A322773, A322774.
%Y Main diagonal is A322775.
%K nonn,tabl
%O 0,4
%A _N. J. A. Sloane_, Dec 30 2018
|
