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

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

Offset

1,2

Comments

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.

Row n contains A001190(n) entries (the Wedderburn-Etherington 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], (18-November-2011)

Emeric Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

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[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].

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[n-1], 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: A047138 A363468 A111299 * A356121 A345326 A327610

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