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A213668 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) consisting of a pair of endvertices joined by n internally disjoint paths of length 2 (the n-ary generalized theta graph THETA_{2,2,...2}; n>=1, 1<=k<=n+2).

%I A213668 #11 Dec 06 2017 03:43:10
%S A213668 1,3,1,0,6,4,1,0,7,10,5,1,0,9,16,15,6,1,0,11,25,30,21,7,1,0,13,36,55,
%T A213668 50,28,8,1,0,15,49,91,105,77,36,9,1,0,17,64,140,196,182,112,45,10,1,0,
%U A213668 19,81,204,336,378,294,156,55,11,1
%N A213668 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) consisting of a pair of endvertices joined by n internally disjoint paths of length 2 (the n-ary generalized theta graph THETA_{2,2,...2}; n>=1, 1<=k<=n+2).
%C A213668 Row n>=2 contains n+1 entries.
%C A213668 Sum of entries in row n=3*2^n-1 = A052940(n) = A153893(n) = A055010(n+1) = A083329(n+1).
%C A213668 The graph G(n) is the join of the graph consisting of 2 isolated vertices and the graph consisting of n isolated vertices. Then the expression of the domination polynomial follows from Theorem 12 of the Akbari et al. reference.
%D A213668 S. Akbari, S. Alikhani, and Y. H. Peng, Characterization of graphs using domination polynomials, European J. Comb., 31, 2010, 1714-1724.
%H A213668 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 A213668 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 A213668 The generating polynomial of row n is p(n)=((1+x)^n-1)*((1+x)^2-1)+x^n+x^2; by definition, p(n) is the domination polynomial of the graph G(n).
%F A213668 Bivariate g.f.: x*z/(1-x*z)-2*x*z/(1-z)+x*z*(1+x)*(2+x)/(1-z-x*z).
%F A213668 T(n,3)=n^2 for n!=3.
%e A213668 Row 1 is 1,3,1 because the graph G(1) is the path abc; there are 1 dominating subset of size 1 ({b}), 3 dominating subsets of size 2 ({a,b}, {a,c}, {b,c}), and 1 dominating subset of size 3 ({a,b,c}).
%e A213668 Row 2 is 0,6,4,1 because the graph G(2) is the cycle a-b-c-d-a and has dominating subsets ab, ac, ad, bc, bd, cd, abc, abd, acd, bcd, and abcd (see A212634).
%e A213668 Triangle starts:
%e A213668 1,3,1;
%e A213668 0,6,4,1;
%e A213668 0,7,10,5,1;
%e A213668 0,9,16,15,6,1;
%p A213668 p := proc (n) options operator, arrow: ((1+x)^n-1)*((1+x)^2-1)+x^n+x^2 end proc: for n to 12 do seq(coeff(p(n), x, k), k = 1 .. n+2) end do; # yields sequence in triangular form
%t A213668 T[n_, k_] := SeriesCoefficient[((1+x)^n-1) ((1+x)^2-1)+x^n+x^2, {x, 0, k}];
%t A213668 Table[T[n, k], {n, 1, 9}, {k, 1, n+2}] // Flatten (* _Jean-François Alcover_, Dec 06 2017 *)
%Y A213668 Cf. A052940, A153893, A055010, A083329, A212634
%K A213668 nonn,tabf
%O A213668 1,2
%A A213668 _Emeric Deutsch_, Jul 06 2012