

A245824


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


6



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



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



KEYWORD

nonn,tabf


AUTHOR



EXTENSIONS

Ordering of terms corrected by Gus Wiseman, Aug 29 2017


STATUS

approved



