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A089246 Triangular array read by rows: a(n, k) is the number of ordered m-tuples of positive integers (x_1, ..., x_m) such that max x_i = n+1-m and there are k ones (0 <= k <= n).

%I A089246 #12 Feb 26 2021 18:42:59
%S A089246 1,0,1,1,0,1,2,2,0,1,5,5,3,0,1,14,15,9,4,0,1,43,50,31,14,5,0,1,144,
%T A089246 180,118,54,20,6,0,1,523,695,481,229,85,27,7,0,1,2048,2869,2081,1035,
%U A089246 395,125,35,8,0,1,8597,12616,9535,4929,1951,629,175,44,9,0,1,38486,58862
%N A089246 Triangular array read by rows: a(n, k) is the number of ordered m-tuples of positive integers (x_1, ..., x_m) such that max x_i = n+1-m and there are k ones (0 <= k <= n).
%C A089246 The row sums are given by A047970 because row n counts the same tuples as the n-th antidiagonal of A047969.
%H A089246 Mathew Englander, <a href="/A089246/a089246.pdf">Comments on A101494 and A089246, and related sequences</a>
%F A089246 From _Mathew Englander_, Feb 25 2021: (Start)
%F A089246 T(n,k) = 0^(n-k) + Sum_{m = k..n-1} C(m,k) * ((n-m)^(m-k) - (n-1-m)^(m-k)).
%F A089246 T(n,k) = Sum_{j = k+1..n-1} C(j,k)*Sum_{i = j..n-1} T(i,j) for 0 <= k < n-1; T(k+1,k)=0 and T(k,k)=1 for k>=0.
%F A089246 G.f. of row n: 1 + Sum_{i = 1..n} (x+n-i)^(i-1)*(x+n-i-1). (End)
%e A089246 a(5, 0) = 14: (5), (4,2), (2,4), (4,3), (3,4), (4,4),
%e A089246 (3,2,2), (2,3,2), (2,2,3), (3,3,2), (3,2,3), (2,3,3), (3,3,3), (2,2,2,2).
%e A089246 a(5, 1) = 15: (4,1), (1,4), (3,3,1), (3,1,3), (1,3,3), 6 permutations of (3,2,1) and 4 permutations of (2,2,2,1).
%e A089246 Triangle starts:
%e A089246                              [0] 1
%e A089246                            [1] 0, 1
%e A089246                           [2] 1, 0, 1
%e A089246                         [3] 2, 2, 0, 1
%e A089246                        [4] 5, 5, 3, 0, 1
%e A089246                     [5] 14, 15, 9, 4, 0, 1
%e A089246                   [6] 43, 50, 31, 14, 5, 0, 1
%e A089246               [7] 144, 180, 118, 54, 20, 6, 0, 1
%e A089246             [8] 523, 695, 481, 229, 85, 27, 7, 0, 1
%e A089246        [9] 2048, 2869, 2081, 1035, 395, 125, 35, 8, 0, 1
%Y A089246 Cf. A047969, A047970, A089302.
%Y A089246 First differences by column of A101494.
%K A089246 nonn,tabl,easy
%O A089246 0,7
%A A089246 _Alford Arnold_, Dec 22 2003
%E A089246 Edited and extended by _David Wasserman_, Sep 07 2005