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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Dec 30 2018
STATUS
approved