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

Author

Emeric Deutsch, Aug 02 2014

Keywords

• nonn
• tabf

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).

Examples

			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;
		

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.

Crossrefs

Cf. A000081, A001190, A007097, A061773, A111299 (the ordered sequence of all numbers appearing in this sequence), A280994.

Programs

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 *)

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].

Extensions

Ordering of terms corrected by Gus Wiseman, Aug 29 2017