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A198322 The Matula-Goebel numbers of the rooted trees that have palindromic Wiener polynomials.

%I A198322 #17 Jun 18 2024 16:19:29
%S A198322 1,2,7,8,56,76,107,147,163,292,454,839,1433,4221,5833,6137,7987,8626,
%T A198322 16216,17059,17128,17764,23438,25672,36812,41203,45952,46428,51768,
%U A198322 60635,83009,86716,86908,88321,91951,93534,94542,99141,100142,108848,120357,124783,133741,136768,137941,140079,142424,145404,145654
%N A198322 The Matula-Goebel numbers of the rooted trees that have palindromic Wiener polynomials.
%C A198322 The Wiener polynomials are assumed to have zero constant terms.
%C A198322 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 A198322 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 A198322 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 A198322 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 A198322 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 A198322 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 A198322 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 A198322 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A198322 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 A198322 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 A198322 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;
%t A198322 r[n_] := FactorInteger[n][[1, 1]];
%t A198322 s[n_] := n/r[n];
%t A198322 R[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, Expand[x*R[PrimePi[n]] + x], True, Expand[R[r[n]] + R[s[n]]]];
%t A198322 W[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, Expand[W[PrimePi[n]] + x*R[PrimePi[n]] + x], True, Expand[W[r[n]] + W[s[n]] + R[r[n]]*R[s[n]]]];
%t A198322 A = {};
%t A198322 Do[If[n == 1 || Expand[x^(1 + Exponent[W[n], x])*(W[n] /. x -> 1/x)] == W[n], Print[n]; A = Union[A, {n}]], {n, 1, 100000}] // Quiet;
%t A198322 A (* _Jean-François Alcover_, Jun 18 2024, after Maple code *)
%Y A198322 Cf. A196059.
%K A198322 nonn
%O A198322 1,2
%A A198322 _Emeric Deutsch_, Oct 24 2011