A213664 - OEIS

%I #8 Dec 12 2013 16:07:15

%S 2,6,4,1,0,7,10,5,1,0,5,16,15,6,1,0,2,18,30,21,7,1,0,0,14,44,50,28,8,

%T 1,0,0,7,48,89,77,36,9,1,0,0,2,39,122,160,112,45,10,1,0,0,0,23,131,

%U 261,265,156,55,11,1,0,0,0,9,110,342,498,413,210,66,12,1

%N Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) obtained by joining a vertex with two consecutive vertices of the cycle graph C_n (n >=3).

%C Row n contain n + 1 entries.

%C Sum of entries in row n = A213665(n).

%H S. Alikhani and E. Deutsch, <a href="http://arxiv.org/abs/1305.3734">Graphs with domination roots in the right half-plane</a>, arXiv preprint arXiv:1305.3734, 2013

%H S. Alikhani and Y. H. Peng, <a href="http://arxiv.org/abs/0905.2251"> Introduction to domination polynomial of a graph</a>, arXiv:0905.2251.

%H T. Kotek, J. Preen, F. Simon, P. Tittmann, and M. Trinks, <a href="http://arxiv.org/abs/1206.5926"> Recurrence relations and splitting formulas for the domination polynomial</a>, arXiv:1206.5926.

%F The generating polynomial g[n] =g[n,x] of row n (= domination polynomial of the graph G(n)) satisfies the recursion relation g[n] = x*g[n-1] + x*g[n-2] + x*g[n-3] (n>=6); g[3]=2x+6x^2+4*x^3+x^4; g[4]=7x^2+10x^3+5x^4+x^5; g[5]=5x^2+16x^3+15x^4+6x^5+x^6.

%e Row 3 is 2,6,4,1 because G(3) is the square abcd with the additional edge bd; all nonempty subsets of {a,b,c,d} are dominating, with the exception of {a} and {c}.

%e Triangle starts:

%e 2,6,4,1;

%e 0,7,10,5,1;

%e 0,5,16,15,6,1;

%e 0,2,18,30,21,7,1;

%p g[3] := 2*x+6*x^2+4*x^3+x^4: g[4] := x^5+5*x^4+10*x^3+7*x^2: g[5] := x^6+6*x^5+15*x^4+16*x^3+5*x^2: for n from 6 to 22 do g[n] := sort(expand(x*g[n-1]+x*g[n-2]+x*g[n-3])) end do: for n from 3 to 14 do seq(coeff(g[n], x, k), k = 1 .. n+1) end do; # yields sequence in triangular form

%Y Cf. A213665

%K nonn,tabf

%O 1,1

%A _Emeric Deutsch_, Jun 30 2012