

A198340


The overall Wiener index of the rooted tree having MatulaGoebel number n.


1



0, 1, 6, 6, 21, 21, 24, 24, 56, 56, 56, 67, 67, 67, 126, 80, 67, 161, 80, 154, 154, 126, 161, 197, 252, 161, 354, 188, 154, 333, 126, 240, 252, 154, 311, 440, 197, 197, 333, 414, 161, 411, 188, 311, 683, 354, 333, 545, 384, 636, 311, 411, 240, 921, 462, 510
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OFFSET

1,3


COMMENTS

The overall Wiener index of any connected graph G is defined as the sum of the Wiener indices of all the subgraphs of G. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.
The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


REFERENCES

F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
D. Bonchev, The overall Wiener index  a new tool for characterization of molecular topology, J. Chem. Inf. Comput. Sci., 2001, 41, 582592.


LINKS

Table of n, a(n) for n=1..56.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

A198339(n) gives the sequence of the MatulaGoebel numbers of all the subtrees of the rooted tree with MatulaGoebel number n. A196051(k) is the Wiener number of the rooted tree with MatulaGoebel number k.


EXAMPLE

a(4)=6 because the rooted tree with MatulaGoebel number 4 is V; each of the 3 onevertex subtrees has Wiener index 0, each of the 2 oneedge subtrees has Wiener index 1, and the tree V itself has Wiener index 4; 0+0+0+1+1+4=6.


MAPLE

m2union := proc (x, y) sort([op(x), op(y)]) end proc: with(numtheory); MRST := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then [1] elif bigomega(n) = 1 then [1, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))] else [seq(seq(mrst[r(n)][i]*mrst[s(n)][j], i = 1 .. nops(mrst[r(n)])), j = 1 .. nops(mrst[s(n)]))] end if end proc: MNRST := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then [] elif bigomega(n) = 1 then m2union(mrst[pi(n)], mnrst[pi(n)]) else m2union(mnrst[r(n)], mnrst[s(n)]) end if end proc: MST := proc (n) m2union(mrst[n], mnrst[n]) end proc: for n to 2000 do mrst[n] := MRST(n): mnrst[n] := MNRST(n): mst[n] := MST(n) end do: W := proc (n) local u, v, E, PL: u := proc (n) options operator, arrow: op(1, factorset(n)) end proc: v := proc (n) options operator, arrow: n/u(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(u(n))+E(v(n)) end if end proc: PL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n))+PL(pi(n)) else PL(u(n))+PL(v(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then W(pi(n))+PL(pi(n))+1+E(pi(n)) else W(u(n))+W(v(n))+PL(u(n))*E(v(n))+PL(v(n))*E(u(n)) end if end proc: OW := proc (n) options operator, arrow: add(W(MST(n)[j]), j = 1 .. nops(MST(n))) end proc: seq(OW(n), n = 1 .. 60);


CROSSREFS

Cf. A196051, A198339.
Sequence in context: A298936 A034695 A339338 * A189980 A188273 A185786
Adjacent sequences: A198337 A198338 A198339 * A198341 A198342 A198343


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Dec 04 2011


STATUS

approved



