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A245824 Triangle read by rows: row n>=1 contains in increasing order the Matula numbers of the rooted binary trees with n leaves.

%I A245824 #28 Aug 29 2017 14:49:24
%S A245824 1,4,14,49,86,301,454,886,1589,1849,3101,3986,6418,13766,9761,13951,
%T A245824 19049,22463,26798,31754,48181,57026,75266,128074,298154,51529,85699,
%U A245824 93793,100561,111139,137987,196249,199591,203878,263431,295969
%N A245824 Triangle read by rows: row n>=1 contains in increasing order the Matula numbers of the rooted binary trees with n leaves.
%C A245824 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 A245824 Row n contains A001190(n) entries (the Wedderburn-Etherington numbers).
%H A245824 Gus Wiseman, <a href="/A245824/b245824.txt">Table of n, a(n) for n = 1..94</a>
%H A245824 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], (18-November-2011)
%H A245824 Emeric Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.
%H A245824 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 A245824 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 A245824 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 A245824 D. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%F A245824 Let H[n] denote the set of binary rooted trees with n leaves or, with some abuse, the set of their Matula numbers (for example, H[1]={1}, H[2]={4}). Each binary rooted tree with n leaves is obtained by identifying the roots of an "elevated" tree from H[k] and of an "elevated" tree from H[n-k] (k=1,..., floor(n/2)). The Maple program is based on this. It makes use of the fact that the Matula number of the "elevation" of a rooted tree with Matula number q has Matula number equal to the q-th prime. The shown program determines H[m] for m=3...9 but shows only H[9].
%e A245824 Row 2 is: 4 (the Matula number of the rooted tree V)
%e A245824 Triangle starts:
%e A245824 1;
%e A245824 4;
%e A245824 14;
%e A245824 49, 86;
%e A245824 301, 454, 886;
%e A245824 1589, 1849, 3101, 3986, 6418, 13766;
%t A245824 nn=9;
%t A245824 allbin[n_]:=allbin[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[allbin/@c]]]/@Select[IntegerPartitions[n-1],Length[#]===2&]];
%t A245824 MGNumber[{}]:=1;MGNumber[x:{__}]:=Times@@Prime/@MGNumber/@x;
%t A245824 Table[Sort[MGNumber/@allbin[n]],{n,1,2nn,2}] (* _Gus Wiseman_, Aug 28 2017 *)
%Y A245824 Cf. A000081, A001190, A007097, A061773, A111299 (the ordered sequence of all numbers appearing in this sequence), A280994.
%K A245824 nonn,tabf
%O A245824 1,2
%A A245824 _Emeric Deutsch_, Aug 02 2014
%E A245824 Ordering of terms corrected by _Gus Wiseman_, Aug 29 2017