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A322770 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.

%I A322770 #35 Jul 21 2021 15:01:38
%S A322770 1,1,1,2,3,5,5,9,18,40,15,31,70,172,457,52,120,299,801,2295,6995,203,
%T A322770 514,1393,4025,12347,40043,136771,877,2407,7023,21709,70843,243235,
%U A322770 875936,3299218,4140,12205,38043,124997,431636,1562071,5908978,23308546,95668354
%N A322770 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 A322770 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. (Background information.)
%H A322770 Alois P. Heinz, <a href="/A322770/b322770.txt">Antidiagonals n = 0..140, flattened</a>
%H A322770 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 A322770 Knuth gives a recurrence using the Bell numbers A000110 (see Maple program).
%e A322770 The array begins:
%e A322770      1,    1,     5,     40,      457,      6995,      136771, ...
%e A322770      1,    3,    18,    172,     2295,     40043,      875936, ...
%e A322770      2,    9,    70,    801,    12347,    243235,     5908978, ...
%e A322770      5,   31,   299,   4025,    70843,   1562071,    41862462, ...
%e A322770     15,  120,  1393,  21709,   431636,  10569612,   310606617, ...
%e A322770     52,  514,  7023, 124997,  2781372,  75114998,  2407527172, ...
%e A322770    203, 2407, 38043, 764538, 18885177, 559057663, 19449364539, ...
%e A322770    ...
%p A322770 B := n -> combinat[bell](n):
%p A322770 Q := proc(m,n) local k; global B; option remember;
%p A322770 if n = 0 then B(m)  else
%p A322770 (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 A322770 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 A322770 Table[Q[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* _Jean-François Alcover_, Jan 02 2019, from Maple *)
%Y A322770 Rows include A094574, A322771, A322772.
%Y A322770 Columns include A000110, A087648, A322773, A322774.
%Y A322770 Main diagonal is A322775.
%K A322770 nonn,tabl
%O A322770 0,4
%A A322770 _N. J. A. Sloane_, Dec 30 2018