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%I #20 Oct 04 2021 13:12:56
%S 1,1,1,1,2,2,1,1,2,2,2,2,2,2,2,1,2,3,1,2,2,2,3,2,3,3,3,2,2,3,2,1,3,2,
%T 2,3,2,2,3,2,3,3,2,2,3,3,3,2,2,3,2,3,1,4,3,2,2,3,2,3,3,3,3,1,3,3,2,2,
%U 3,3,2,3,3,3,3,2,3,4,2,2,4,3,3,3,3,3,3,2
%N The domination number of the rooted tree with Matula-Goebel number n.
%C The domination number of a simple graph G is the minimum cardinality of a dominating subset of G.
%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.
%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 [math.CO], 2009.
%H É. Czabarka, L. Székely, and S. Wagner, <a href="http://dx.doi.org/10.1016/j.dam.2009.07.004">The inverse problem for certain tree parameters</a>, Discrete Appl. Math., 157, 2009, 3314-3319.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F In A212630 one gives the domination polynomial P(n)=P(n,x) of the rooted tree with Matula-Goebel number n. We have a(n) = least exponent in P(n,x).
%e a(5)=2 because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C; {A,B} is a dominating subset and there is no dominating subset of smaller cardinality.
%p with(numtheory): P := proc (n) local r, s, A, B, C:
%p r := n -> op(1, factorset(n)): s := n-> n/r(n):
%p 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:
%p 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:
%p 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:
%p sort(expand(A(n)+B(n))) end proc:
%p A212632 := n->degree(P(n))-degree(numer(subs(x = 1/x, P(n)))): seq(A212632(n), n = 1 .. 120);
%t A[n_] := Which[n == 1, x, PrimeOmega[n] == 1, x*(A[PrimePi[n]] + B[PrimePi[n]] + c[PrimePi[n]]), True, A[r[n]]*A[s[n]]/x];
%t B[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, A[PrimePi[n]], True, Expand[B[r[n]]*B[s[n]] + B[r[n]]*c[s[n]] + B[s[n]]*c[r[n]]]];
%t c[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, B[PrimePi[n]], True, Expand[c[r[n]]*c[s[n]]]];
%t r[n_] := FactorInteger[n][[1, 1]];
%t s[n_] := n/r[n];
%t P[n_] := Expand[A[n] + B[n]];
%t a[n_] := Exponent[P[n], x] - Exponent[Numerator[P[n] /. x -> 1/x // Together], x];
%t Array[a, 100] (* _Jean-François Alcover_, Nov 14 2017, after _Emeric Deutsch_ *)
%Y Cf. A212618 - A212631.
%K nonn
%O 1,5
%A _Emeric Deutsch_, Jun 11 2012
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