The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #13 Aug 30 2019 14:49:31
%S 1,1,2,1,4,5,1,7,14,15,1,11,36,51,52,1,16,81,171,202,203,1,22,162,512,
%T 813,876,877,1,29,295,1345,3046,4012,4139,4140,1,37,499,3145,10096,
%U 17866,20891,21146,21147,1,46,796,6676,29503,72028,106133,115463,115974,115975
%N T(n,k) = number of nonnegative integer arrays of length n+k-1 with new values 0 upwards introduced in order, and containing the value k-1.
%C Table starts
%C ....1.....1......1......1.......1........1........1.........1..........1
%C ....2.....4......7.....11......16.......22.......29........37.........46
%C ....5....14.....36.....81.....162......295......499.......796.......1211
%C ...15....51....171....512....1345.....3145.....6676.....13091......24047
%C ...52...202....813...3046...10096....29503....77078....183074.....401337
%C ..203...876...4012..17866...72028...256565...810470...2300949....5957407
%C ..877..4139..20891.106133..503295..2134122..8016373..26869727...81381744
%C .4140.21146.115463.649045.3513522.17337685.76199007.298009584.1046405027
%C Reading along antidiagonals seems to create A137650. - _R. J. Mathar_, Nov 29 2015
%C See also A133611. - _Alois P. Heinz_, Aug 30 2019
%H R. H. Hardin, <a href="/A211561/b211561.txt">Table of n, a(n) for n = 1..9999</a>
%F Empirical: T(n,k) = Sum_{j=k..n+k-1} stirling2(n+k-1,j)
%e Some solutions for n=5, k=4:
%e ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e ..1....1....1....0....1....1....1....1....0....1....1....1....1....1....1....0
%e ..1....2....2....0....0....2....2....0....1....2....2....2....2....0....2....1
%e ..2....0....2....0....2....0....3....2....2....2....3....3....2....2....0....2
%e ..3....1....3....1....3....2....1....3....3....2....1....3....3....2....1....2
%e ..4....0....3....0....3....3....4....1....3....3....0....2....4....3....2....2
%e ..5....3....3....2....4....4....2....1....2....2....1....0....4....3....3....2
%e ..2....0....1....3....5....4....4....4....4....2....0....4....3....1....2....3
%Y Column 1 is A000110.
%Y Column 2 is A058692(n+1).
%Y Column 3 is A058681(n+1).
%Y Row 2 is A000124.
%Y Cf. A133611, A137650.
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_, Apr 15 2012