%I #11 Dec 02 2017 03:04:28
%S 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,
%T 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,
%U 253,263,156,55,11,1,0,0,0,7,96,322,488,411,210,66,12,1
%N 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).
%C 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.
%C 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).
%C Sum of entries in row n is A213655(n).
%D S. Alikhani and Y. H. Peng, Dominating sets and domination polynomials of certain graphs, II, Opuscula Math., 30, No. 1, 2010, 37-51.
%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.
%F 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.
%e T(1,1)=2 because in the theta-graph TH(2,2,1) any of the two vertices of degree 3 is dominating.
%e Irregular triangle starts:
%e 2,6,4,1;
%e 0,7,10,5,1;
%e 0,3,16,15,6,1;
%e 0,2,16,30,21,7,1;
%p 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
%t 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])]];
%t Table[CoefficientList[p[n], x] // Rest, {n, 1, 13}] // Flatten (* _Jean-François Alcover_, Dec 02 2017, from Maple *)
%Y Cf. A213655
%K nonn,tabf
%O 1,1
%A _Emeric Deutsch_, Jun 18 2012