OFFSET
1,2
COMMENTS
Row n>=2 contains n+1 entries.
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.
REFERENCES
S. Akbari, S. Alikhani, and Y. H. Peng, Characterization of graphs using domination polynomials, European J. Comb., 31, 2010, 1714-1724.
LINKS
S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251.
T. Kotek, J. Preen, F. Simon, P. Tittmann, and M. Trinks, Recurrence relations and splitting formulas for the domination polynomial, arXiv:1206.5926.
FORMULA
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).
Bivariate g.f.: x*z/(1-x*z)-2*x*z/(1-z)+x*z*(1+x)*(2+x)/(1-z-x*z).
T(n,3)=n^2 for n!=3.
EXAMPLE
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}).
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).
Triangle starts:
1,3,1;
0,6,4,1;
0,7,10,5,1;
0,9,16,15,6,1;
MAPLE
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
MATHEMATICA
T[n_, k_] := SeriesCoefficient[((1+x)^n-1) ((1+x)^2-1)+x^n+x^2, {x, 0, k}];
Table[T[n, k], {n, 1, 9}, {k, 1, n+2}] // Flatten (* Jean-François Alcover, Dec 06 2017 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jul 06 2012
STATUS
approved