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A198338 Irregular triangle read by rows: row n is the sequence of Matula numbers of the rooted subtrees of the rooted tree with Matula-Goebel number n. A root subtree of a rooted tree T is a subtree of T containing the root.

%I A198338 #9 Jan 07 2013 13:05:55
%S A198338 1,1,2,1,2,3,1,2,2,4,1,2,3,5,1,2,2,3,4,6,1,2,3,3,7,1,2,2,2,4,4,4,8,1,
%T A198338 2,2,3,3,4,6,6,9,1,2,2,3,4,5,6,10,1,2,3,5,11,1,2,2,2,3,4,4,4,6,6,8,12,
%U A198338 1,2,3,3,4,6,6,7,15,1,2,2,3,3,4,5,6,6
%N A198338 Irregular triangle read by rows: row n is the sequence of Matula numbers of the rooted subtrees of the rooted tree with Matula-Goebel number n. A root subtree of a rooted tree T is a subtree of T containing the root.
%C A198338 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.
%C A198338 Number of entries in row n is A184160(n). Row n>=2 can be easily identified: it starts with the entry following the first occurrence of n-1 and it ends with the first occurrence of n.
%D A198338 F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A198338 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A198338 I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A198338 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A198338 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288
%F A198338 Row 1 is [1]; if n = p(t) (= the t-th prime), then row n is [1, p(a), p(b), ... ], where [a,b,...] is row t; if n=rs (r,s >=2), then row n consists of the numbers r[i]*s[j], where [r[1], r[2],...] is row r and [s[1], s[2], ...] is row s. The Maple program, based on this recursive procedure, yields row n (<=1000; adjustable) with the command MRST(n).
%e A198338 Row 4 is [1,2,2,4] because the rooted tree with Matula-Goebel number 4 is V and its root subtrees are *, |, |, and V.
%e A198338 Triangle starts:
%e A198338 1;
%e A198338 1,2;
%e A198338 1,2,3;
%e A198338 1,2,2,4;
%e A198338 1,2,3,5;
%e A198338 1,2,2,3,4,6;
%p A198338 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 sort([1, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))]) else sort([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: for n to 1000 do mrst[n] := MRST(n) end do;
%Y A198338 Cf. A198339.
%K A198338 nonn,tabf
%O A198338 1,3
%A A198338 _Emeric Deutsch_, Dec 04 2011