%I #12 Mar 20 2013 12:33:35

%S 1,1,0,1,1,0,1,2,0,0,1,3,2,0,0,1,4,5,0,0,0,1,5,9,5,0,0,0,1,6,14,14,0,

%T 0,0,0,0,7,20,28,14,0,0,0,0,0,7,27,48,42,0,0,0,0,0,0,0,34,75,90,42,0,

%U 0,0,0,0,0,0,34,109,165,132,0,0,0,0,0,0,0,0,0,143,274,297,132,0,0,0,0,0,0,0,0,0,143,417,571,429,0,0,0,0,0,0,0

%N Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 8, T(0,k)= 1 if 0<=k<=7, T(n,k) = T(n-1,k) + T(n,k-1).

%C A hexagon arithmetic of E. Lucas.

%F T(n,n) = A080938(n).

%F T(n,n+1) = A080938(n+1).

%F T(n,n+2) = A094826(n+1).

%F T(n,n+3) = A094827(n+1).

%F T(n,n+4) = A094828(n+2).

%F T(n,n+5) = A094829(n+2).

%F T(n,n+6) = T(n,n+7) = A094256(n+1).

%F Sum_{k, 0<=k<=n} T(n-k,k) = A061551(n).

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ... row n=0

%e 0, 1, 2, 3, 4, 5, 6, 7, 7, 0, 0, 0, 0, 0, 0, ... row n=1

%e 0, 0, 2, 5, 9, 14, 20, 27, 34, 34, 0, 0, 0, ... row n=2

%e 0, 0, 0, 5, 14, 28, 48, 75, 109, 143, 143, 0, 0, ... row n=3

%e 0, 0, 0, 0, 14, 42, 90, 165, 274, 417, 560, 560, 0, ... row n=4

%e 0, 0, 0, 0, 0, 42, 132, 297, 571, 988, 1548, 2108, 2108, 0, ... row n=5

%e ...

%t t[0, k_ /; k <= 7] = 1; t[n_, k_] /; k < n || k > n+7 = 0; t[n_, k_] := t[n, k] = t[n-1, k] + t[n, k-1]; Table[t[n-k, k], {n, 0, 13}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Mar 18 2013 *)

%Y Cf. Similar sequence: A216230, A216228, A216226, A216238, A216054, A217257.

%K nonn,tabl

%O 0,8

%A _Philippe Deléham_, Mar 17 2013