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A196057 Number of sibling pairs in the rooted tree with Matula-Goebel number n. 1
0, 0, 0, 1, 0, 1, 1, 3, 1, 1, 0, 3, 1, 2, 1, 6, 1, 3, 3, 3, 2, 1, 1, 6, 1, 2, 3, 4, 1, 3, 0, 10, 1, 2, 2, 6, 3, 4, 2, 6, 1, 4, 2, 3, 3, 2, 1, 10, 3, 3, 2, 4, 6, 6, 1, 7, 4, 2, 1, 6, 3, 1, 4, 15, 2, 3, 3, 4, 2, 4, 3, 10, 2, 4, 3, 6, 2, 4, 1, 10, 6, 2, 1, 7, 2, 3, 2, 6, 6, 6, 3, 4, 1, 2, 4, 15, 1, 5, 3, 6, 2, 4, 3, 7, 4, 7, 4, 10, 1, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

A sibling pair is an unordered pair of nodes having the same parent.

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.

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 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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011

Index entries for sequences related to Matula-Goebel numbers

FORMULA

a(1)=0; if n=p(t) (=the t-th prime), then a(n)=a(t); if n=rs (r,s,>=2), then a(n)=a(r)+a(s)+G(r)G(s), where G(m) is the number of prime divisors of m, counted with multiplicities. The Maple program is based on this recursive formula.

EXAMPLE

a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.

a(2^m) = binom(m,2) because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

MAPLE

with(numtheory): a := proc (n) local u, v: u := proc (n) options operator, arrow: op(1, factorset(n)) end proc: v := proc (n) options operator, arrow: n/u(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n)) else a(u(n))+a(v(n))+bigomega(u(n))*bigomega(v(n)) end if end proc: seq(a(n), n = 1 .. 110);

PROG

(Haskell)

import Data.List (genericIndex)

a196057 n = genericIndex a196057_list (n - 1)

a196057_list = 0 : g 2 where

   g x = y : g (x + 1) where

     y | t > 0     = a196057 t

       | otherwise = a196057 r + a196057 s + a001222 r * a001222 s

       where t = a049084 x; r = a020639 x; s = x `div` r

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS

Cf. A049084, A020639, A001222.

Sequence in context: A220419 A085565 A216677 * A058395 A035694 A006941

Adjacent sequences:  A196054 A196055 A196056 * A196058 A196059 A196060

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 30 2011

STATUS

approved

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Last modified January 17 18:21 EST 2019. Contains 319250 sequences. (Running on oeis4.)