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%I #21 Nov 03 2019 19:45:21
%S 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,
%T 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,
%U 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
%N Number of sibling pairs in the rooted tree with Matula-Goebel number n.
%C A sibling pair is an unordered pair of nodes having the same parent.
%C 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.
%H Reinhard Zumkeller, <a href="/A196057/b196057.txt">Table of n, a(n) for n = 1..10000</a>
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%H 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 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 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 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 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F 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.
%e a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
%e a(2^m) = binomial(m,2) because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%p 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);
%o (Haskell)
%o import Data.List (genericIndex)
%o a196057 n = genericIndex a196057_list (n - 1)
%o a196057_list = 0 : g 2 where
%o g x = y : g (x + 1) where
%o y | t > 0 = a196057 t
%o | otherwise = a196057 r + a196057 s + a001222 r * a001222 s
%o where t = a049084 x; r = a020639 x; s = x `div` r
%o -- _Reinhard Zumkeller_, Sep 03 2013
%Y Cf. A049084, A020639, A001222.
%K nonn
%O 1,8
%A _Emeric Deutsch_, Sep 30 2011
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