login
Array: T(n,k)=k-n+1 for k>=n; T(n,k)=0 for k<n; by antidiagonals.
3

%I #7 Jul 22 2017 17:28:38

%S 1,2,0,3,1,0,4,2,0,0,5,3,1,0,0,6,4,2,0,0,0,7,5,3,1,0,0,0,8,6,4,2,0,0,

%T 0,0,9,7,5,3,1,0,0,0,0,10,8,6,4,2,0,0,0,0,0,11,9,7,5,3,1,0,0,0,0,0,12,

%U 10,8,6,4,2,0,0,0,0,0,0,13,11,9,7,5,3,1,0,0,0,0,0,0,14,12,10,8,6,4,2,0,0,0,0,0,0,0

%N Array: T(n,k)=k-n+1 for k>=n; T(n,k)=0 for k<n; by antidiagonals.

%C A member of the accumulation chain

%C ...< A185916 < A185914 < A185915 < ...

%C (See A144112 for definitions of weight array and accumulation array.)

%H G. C. Greubel, <a href="/A185914/b185914.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F T(n,k) = k-n+1 for k>=n; T(n,k)=0 for k<n; k>=1, n>=1.

%e Northwest corner:

%e 1...2...3...4...5...6...7...8...9

%e 0...1...2...3...4...5...6...7...8

%e 0...0...1...2...3...4...5...6...7

%e 0...0...0...1...2...3...4...5...6

%e 0...0...0...0...1...2...3...4...5

%t (* This program generates the array A185914, its accumulation array A185915, and its weight array A185916. *)

%t f[n_,0]:=0;f[0,k_]:=0; (* needed for the weight array *)

%t f[n_,k_]:=k-n+1; f[n_,k_]:=0/;k<n;

%t TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185914 *)

%t Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

%t s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}];

%t TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A184915 *)

%t Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

%t w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];

%t TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* A184916 *)

%t Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

%Y Cf. A144112, A185915, A185916.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 06 2011