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

1, 2, 7, 8, 56, 76, 107, 147, 163, 292, 454, 839, 1433, 4221, 5833, 6137, 7987, 8626, 16216, 17059, 17128, 17764, 23438, 25672, 36812, 41203, 45952, 46428, 51768, 60635, 83009, 86716, 86908, 88321, 91951, 93534, 94542, 99141, 100142, 108848, 120357, 124783, 133741, 136768, 137941, 140079, 142424, 145404, 145654

Offset: 1

Author

Emeric Deutsch, Oct 24 2011

Keywords

• nonn

Comments

The Wiener polynomials are assumed to have zero constant terms.

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.

Examples

			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.
		

References

• 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.

Links

• F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
• I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
• I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
• D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A196059.

Programs

Maple

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;

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
R[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, Expand[x*R[PrimePi[n]] + x], True, Expand[R[r[n]] + R[s[n]]]];
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]]]];
A = {};
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;
A (* Jean-François Alcover, Jun 18 2024, after Maple code *)

Formula

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).