

A245824


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


3



1, 4, 14, 49, 86, 301, 454, 886, 1589, 1849, 3101, 3986, 6418, 13766, 9761, 13951, 19049, 22463, 26798, 31754, 48181, 57026, 75266, 128074, 298154, 51529, 85699, 93793, 100561, 111139, 137987, 196249, 199591, 203878, 263431, 295969
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OFFSET

1,2


COMMENTS

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.
Row n contains A001190(n) entries (the WedderburnEtherington numbers).


LINKS

Gus Wiseman, Table of n, a(n) for n = 1..94
Emeric Deutsch, Tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], (18November2011)
Emeric Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 23142322.
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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.


FORMULA

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[nk] (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 qth prime. The shown program determines H[m] for m=3...9 but shows only H[9].


EXAMPLE

Row 2 is: 4 (the Matula number of the rooted tree V)
Triangle starts:
1;
4;
14;
49, 86;
301, 454, 886;
1589, 1849, 3101, 3986, 6418, 13766;


MATHEMATICA

nn=9;
allbin[n_]:=allbin[n]=If[n===1, {{}}, Join@@Function[c, Union[Sort/@Tuples[allbin/@c]]]/@Select[IntegerPartitions[n1], Length[#]===2&]];
MGNumber[{}]:=1; MGNumber[x:{__}]:=Times@@Prime/@MGNumber/@x;
Table[Sort[MGNumber/@allbin[n]], {n, 1, 2nn, 2}] (* Gus Wiseman, Aug 28 2017 *)


CROSSREFS

Cf. A000081, A001190, A007097, A061773, A111299 (the ordered sequence of all numbers appearing in this sequence), A280994.
Sequence in context: A220819 A047138 A111299 * A110686 A071729 A278692
Adjacent sequences: A245821 A245822 A245823 * A245825 A245826 A245827


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Aug 02 2014


EXTENSIONS

Ordering of terms corrected by Gus Wiseman, Aug 29 2017


STATUS

approved



