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.

1, 1, 1, 2, 3, 5, 5, 9, 18, 40, 15, 31, 70, 172, 457, 52, 120, 299, 801, 2295, 6995, 203, 514, 1393, 4025, 12347, 40043, 136771, 877, 2407, 7023, 21709, 70843, 243235, 875936, 3299218, 4140, 12205, 38043, 124997, 431636, 1562071, 5908978, 23308546, 95668354

Offset

0,4

References

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. (Background information.)

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

D. E. Knuth, Partitioning a multiset into submultisets, Email to N. J. A. Sloane, Dec 29 2018.

Formula

Knuth gives a recurrence using the Bell numbers A000110 (see Maple program).

Example

The array begins:
     1,    1,     5,     40,      457,      6995,      136771, ...
     1,    3,    18,    172,     2295,     40043,      875936, ...
     2,    9,    70,    801,    12347,    243235,     5908978, ...
     5,   31,   299,   4025,    70843,   1562071,    41862462, ...
    15,  120,  1393,  21709,   431636,  10569612,   310606617, ...
    52,  514,  7023, 124997,  2781372,  75114998,  2407527172, ...
   203, 2407, 38043, 764538, 18885177, 559057663, 19449364539, ...
   ...

Maple

B := n -> combinat[bell](n):
Q := proc(m, n) local k; global B; option remember;
if n = 0 then B(m)  else
(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)

Mathematica

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}])];
Table[Q[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 02 2019, from Maple *)

Crossrefs

Rows include A094574, A322771, A322772.

Columns include A000110, A087648, A322773, A322774.

Main diagonal is A322775.

Sequence in context: A222705 A241381 A237365 * A257008 A338750 A265822

Adjacent sequences: A322767 A322768 A322769 * A322771 A322772 A322773

Keyword

nonn,tabl

Author

N. J. A. Sloane, Dec 30 2018

Status

approved