This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A202852 #9 Oct 23 2021 21:18:07
%S A202852 343,908,1029,1421,1813,2270,2724,2891,3087,3209,3412,3773,3859,4263,
%T A202852 4459,4618,4753,4948,5439,5537,5675,5887,6548,6810,7399,7511
%N A202852 Matula-Goebel numbers of rooted trees with no perfect matching and such that 2 is an eigenvalue of the Laplacian matrix.
%C A202852 It is known that 2 is an eigenvalue of the Laplacian of any tree with a perfect matching (see the Ming & Zhang reference, Theorem 2).
%C A202852 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.
%D A202852 F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A202852 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A202852 I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A202852 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D A202852 Guo Ji Ming and Tan Shang Wang, A relation between the matching number and Laplacian spectrum of a graph, Linear Algebra and its Appl., 325, 2001, 71-74.
%D A202852 Yi-zheng Fan, On the eigenvalue two and matching number of a tree, Acta Math. Appl. Sinica, English Series, 20, 2004, 257-262.
%H A202852 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288.
%H A202852 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A202852 Set {A193402(n), n>=1} minus set {A193405(n), n>=1}.
%e A202852 The numbers 343, 908, and 3209 are in the sequence; they are the rooted trees obtained from the tree of Fig. 2 in the Fan reference by taking the root at different vertices. The tree has no perfect matching because it has 2 leaves with the same parent. Its Laplacian matrix has characteristic polynomial x(x-2)(x-5)(x-1)^3*(x^2 - 4x + 1)^2.
%Y A202852 Cf. A193402, A193405
%K A202852 nonn
%O A202852 1,1
%A A202852 _Emeric Deutsch_, Feb 13 2012