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A187152 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.

%I A187152 #19 Nov 23 2019 04:03:51
%S A187152 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 A187152 103,331,1064,2356,3,10,32,103,331,1064,3420,7573,3,10,32,103,331,
%U A187152 1064,3420,10993,24342,3,10,32,103,331,1064,3420,10993,35335,78243
%N A187152 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 A187152 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 A187152 For m >= n:
%F A187152 T(m,n) =
%F A187152    A030186(m)   if m = n
%F A187152    A033505(n+1) if m >= n+1
%F A187152 Symmetrically extended by T(n,m)=T(m,n).
%F A187152 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 A187152 2;
%e A187152 3 7;
%e A187152 3 10 22;
%e A187152 3 10 32 71;
%e A187152 3 10 32 103 228;
%e A187152 3 10 32 103 331 733;
%e A187152 3 10 32 103 331 1064 2356;
%e A187152 3 10 32 103 331 1064 3420 7573;
%e A187152 3 10 32 103 331 1064 3420 10993 24342;
%e A187152 3 10 32 103 331 1064 3420 10993 35335 78243;
%e A187152 3 10 32 103 331 1064 3420 10993 35335 113578 251498;
%Y A187152 Cf. A033505, A030186.
%K A187152 nonn,tabl
%O A187152 1,1
%A A187152 _Steffen Eger_, Mar 06 2011