A196061 The multiplicative Wiener index of the rooted tree with Matula-Goebel number n.

1, 2, 2, 12, 12, 8, 8, 288, 288, 288, 144, 144, 144, 34560, 64, 144, 10368, 64, 13824, 13824, 34560, 10368, 3456, 24883200, 10368, 2985984, 5184, 13824, 4976640, 34560, 1024, 24883200, 13824, 8294400, 746496, 3456, 3456, 4976640, 1327104, 10368, 1492992, 5184, 8294400, 7166361600

Offset: 2

Author

Emeric Deutsch, Sep 30 2011

Keywords

• nonn

Comments

The multiplicative Wiener index of a connected graph is the product of the distances between all unordered pairs of vertices in the graph.

The Matula-Goebel number of a rooted tree is 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

			a(7)=8 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with distances 1,1,1,2,2,2; product of distances is 8.
a(2^m) = 2^[m(m-1)/2] because the rooted tree with Matula-Goebel number 2^m is a star with m edges and we have m distances 1 and m(m-1)/2 distances 2.
		

References

• 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, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.
• I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, On the multiplicative Wiener index and its possible chemical applications, Monatshefte f. Chemie, 131, 2000, 421-427.
• 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 Review, 10, 1968, 273.

Links

• Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
• 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 := proc (n) options operator, arrow: product(k^coeff(W(n), x, k), k = 1 .. degree(W(n))) end proc: seq(a(n), n = 2 .. 45);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
R[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*R[PrimePi[n]] + x, True, R[r[n]] + R[s[n]]];
W[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, W[PrimePi[n]] + x*R[PrimePi[n]] + x, True, W[r[n]] + W[s[n]] + R[r[n]]*R[s[n]]];
a[n_] := Product[k^Coefficient[W[n], x, k], {k, 1, Exponent[W[n], x]}];
Table[a[n], {n, 2, 45}] (* Jean-François Alcover, Jun 22 2024, after Maple code *)

Formula

a(n) = Product_{k=1..d} k^c(k), where d is the diameter of the rooted tree with Matula-Goebel number n, and c(k) is the number of pairs of nodes at distance k (all these data are contained in the Wiener polynomial; see A196059). The Maple program is based on the above.