A213654 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with cardinality k of the theta-graph TH(2,2,n) (n>=1, 1<=k<=n+3).

2, 6, 4, 1, 0, 7, 10, 5, 1, 0, 3, 16, 15, 6, 1, 0, 2, 16, 30, 21, 7, 1, 0, 0, 12, 42, 50, 28, 8, 1, 0, 0, 5, 44, 87, 77, 36, 9, 1, 0, 0, 2, 33, 116, 158, 112, 45, 10, 1, 0, 0, 0, 19, 119, 253, 263, 156, 55, 11, 1, 0, 0, 0, 7, 96, 322, 488, 411, 210, 66, 12, 1

Offset: 1

Author

Emeric Deutsch, Jun 18 2012

Keywords

• nonn
• tabf

Comments

A theta-graph is a graph consisting of two vertices of degree three, connected by three paths of one or more edges each. In the theta-graph TH(2,2,n) the three paths have 2, 2, and n edges, respectively.

The entries in row n are the coefficients of the domination polynomial of the theta-graph TH(2,2,n) (see the Alikhani and Peng arxiv reference).

Sum of entries in row n is A213655(n).

Examples

			T(1,1)=2 because in the theta-graph TH(2,2,1) any of the two vertices of degree 3 is dominating.
Irregular triangle starts:
2,6,4,1;
0,7,10,5,1;
0,3,16,15,6,1;
0,2,16,30,21,7,1;
		

References

• S. Alikhani and Y. H. Peng, Dominating sets and domination polynomials of certain graphs, II, Opuscula Math., 30, No. 1, 2010, 37-51.

Links

• S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251.

Crossrefs

Cf. A213655

Programs

Maple

p:=proc(n) if n = 1 then sort(x^4+4*x^3+6*x^2+2*x) elif n = 2 then sort(x^5+5*x^4+10*x^3+7*x^2) elif n = 3 then sort(x^6+6*x^5+15*x^4+16*x^3+3*x^2) else sort(expand(x*(p(n-1)+p(n-2)+p(n-3)))) end if end proc: for n to 13 do seq(coeff(p(n), x, k), k = 1 .. n+3) end do; # yields sequence in triangular form

Mathematica

p[n_] := p[n] = Switch[n, 1, x^4 + 4*x^3 + 6*x^2 + 2*x, 2, x^5 + 5*x^4 + 10*x^3 + 7*x^2, 3, x^6 + 6*x^5 + 15*x^4 + 16*x^3 + 3*x^2, _, Expand[x* (p[n - 1] + p[n - 2] + p[n - 3])]];
Table[CoefficientList[p[n], x] // Rest, {n, 1, 13}] // Flatten (* Jean-François Alcover, Dec 02 2017, from Maple *)

Formula

If p(n)=p(n,x) denotes the generating polynomial of row n (called the domination polynomial of the theta-graph TH(2,2,n), then p(n) = x*[p(n-1) + p(n-2) + p(n-3)] for n>=4; p(1), p(2), p(3) are given in the Maple program.