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A211561 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.

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