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%I #13 Mar 30 2020 08:33:05
%S 1,2,7,8,56,76,107,147,163,292,454,839,1433,4221,5833,6137,7987,8626,
%T 16216,17059,17128,17764,23438,25672,36812,41203,45952,46428,51768,
%U 60635,83009,86716,86908,88321,91951,93534,94542,99141,100142,108848,120357,124783,133741,136768,137941,140079,142424,145404,145654
%N The Matula-Goebel numbers of the rooted trees that have palindromic Wiener polynomials.
%C The Wiener polynomials are assumed to have zero constant terms.
%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.
%D G. Caporossi, A. A. Dobrynin, I. Gutman, and P. Hansen, Trees with palindromic Hosoya polynomials, Graph Theory Notes of New York, XXXVI, 1999, 10-16.
%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 B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F The Wiener polynomial W(n,x) of the rooted tree corresponding to the Matula-Goebel number n is given in A196059. It is palindromic if and only if x^{1+degree(W(n,x))}*W(n,1/x)=W(n,x).
%e 7 is in the sequence because the rooted tree with Matula-Goebel number 7 is Y; 3 distances are equal to 1 and 3 distances are equal to 2; Wiener polynomial is 3x+3x^2.
%p with(numtheory): W := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(W(pi(n))+x*R(pi(n))+x)) else sort(expand(W(r(n))+W(s(n))+R(r(n))*R(s(n)))) end if end proc: A := {}: for n to 100000 do if expand(x^(1+degree(W(n)))*subs(x = 1/x, W(n))) = W(n) then A := `union`(A, {n}) else end if end do: A;
%Y Cf. A196059.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Oct 24 2011
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