%N Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the rooted tree with Matula-Goebel number n (n>=1, k>=1).
%C The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
%C The entries in row n are the coefficients of the domination polynomial of the rooted tree with Matula-Goebel number n (see the Alikhani and Peng reference).
%C Sum of entries in row n = A212631(n) (number of dominating subsets).
%C The order of the first nonzero entry in row n = A212632(n) (the domination number).
%D F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
%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 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F Let A(n)=A(n,x), B(n)=B(n,x), and C(n)=C(n,x) be the generating polynomial with respect to size of the dominating subsets which contain the root, of the dominating subsets which do not contain the root, and of the subsets which dominate all vertices except the root, respectively, of the rooted tree with Matula-Goebel number n. We have A(1)=x, B(1)=0, C(1)=1, A(t-th prime) = x [A(t)+B(t)+C(t)], B(t-th prime) = A(t), C(t-th prime) = B(t); A(rs) = A(r)A(s)/x, B(rs) = B(r)B(s) + B(r)C(s) + B(s)C(r) (r,s>=2). The generating polynomial of the dominating subsets with respect to size (i.e. the domination polynomial) is P(n)=P(n,x)=A(n)+B(n). The Maple program is based on these recurrence relations.
%e Row 3 is [1,3,1] because the rooted tree with Matula-Goebel number 3 is the path tree R - A - B; it has 1, 3, and 1 dominating subsets with 1, 2, and 3 vertices, respectively: [A], [RA, RB, AB], and [RAB].
%p with(numtheory): P := proc (n) local r, s, A, B, C: r := n-> op(1, factorset(n)): s := n-> n/r(n): A := proc (n) if n = 1 then x elif bigomega(n) = 1 then x*(A(pi(n))+B(pi(n))+C(pi(n))) else A(r(n))*A(s(n))/x end if end proc: B := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then A(pi(n)) else sort(expand(B(r(n))*B(s(n))+B(r(n))*C(s(n))+B(s(n))*C(r(n)))) end if end proc: C := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then B(pi(n)) else expand(C(r(n))*C(s(n))) end if end proc: sort(expand(A(n)+B(n))) end proc: for n to 20 do seq(coeff(P(n), x, j), j = 1 .. degree(P(n))) end do; # yields sequence in triangular form
%Y Cf. A212618 - A212632.
%A _Emeric Deutsch_, Jun 11 2012