%I #19 Nov 23 2019 04:03:51

%S 2,3,7,3,10,22,3,10,32,71,3,10,32,103,228,3,10,32,103,331,733,3,10,32,

%T 103,331,1064,2356,3,10,32,103,331,1064,3420,7573,3,10,32,103,331,

%U 1064,3420,10993,24342,3,10,32,103,331,1064,3420,10993,35335,78243

%N Triangle T(m,n), read by rows: Number of bipartite labeled graphs (V,E) with vertices A={a_1,...,a_m} and B={b_1,...,b_n} where for any vertex in V at most one edge in E is allowed. Additionally, an edge {a_k,b_l} is allowed only when |k-l|<=1.

%C This also has the obvious corresponding string alignment interpretation where we allow only one-to-one alignments between strings a_1...a_m and b_1...b_n, and additionally demand that aligned characters have a distance of at most 1.

%F For m >= n:

%F T(m,n) =

%F A030186(m) if m = n

%F A033505(n+1) if m >= n+1

%F Symmetrically extended by T(n,m)=T(m,n).

%F Both the diagonal and the off-diagonals follow the recurrence a(n) = 3*a(n-1) + a(n-2) - a(n-3), n >= 3, with different initial conditions; 2,7,22 and 3,10,32, respectively.

%e 2;

%e 3 7;

%e 3 10 22;

%e 3 10 32 71;

%e 3 10 32 103 228;

%e 3 10 32 103 331 733;

%e 3 10 32 103 331 1064 2356;

%e 3 10 32 103 331 1064 3420 7573;

%e 3 10 32 103 331 1064 3420 10993 24342;

%e 3 10 32 103 331 1064 3420 10993 35335 78243;

%e 3 10 32 103 331 1064 3420 10993 35335 113578 251498;

%Y Cf. A033505, A030186.

%K nonn,tabl

%O 1,1

%A _Steffen Eger_, Mar 06 2011