

A202854


The MatulaGöbel numbers of rooted trees T for which the sequence formed by the number of kmatchings of T (k=0,1,2,...) is palindromic.


1



1, 2, 5, 6, 18, 23, 26, 41, 54, 78, 103, 122, 162, 167, 202, 234, 283, 338, 366, 419, 486, 502, 547, 606, 643, 702, 794, 1009, 1014, 1093, 1098, 1346, 1458, 1506, 1543, 1586, 1597, 1818, 1906, 1999, 2106, 2371, 2382, 2462, 2626, 2719, 2962
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OFFSET

1,2


COMMENTS

Alternatively, the MatulaGöbel numbers of rooted trees for which the matchinggenerating polynomial is palindromic.
A kmatching in a graph is a set of k edges, no two of which have a vertex in common.
The MatulaGöbel 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 MatulaGöbel numbers of the m branches of T.
After activating the Maple program, the command m(n) will yield the matchinggenerating polynomial of the rooted tree having MatulaGöbel number n.
The given Maple program gives the required MatulaGöbel numbers up to L=200 (adjustable).


REFERENCES

C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.


LINKS

Table of n, a(n) for n=1..47.
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 33143319.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
F. Göbel, 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.
Eric Weisstein's World of Mathematics, MatchingGenerating Polynomial
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Define b(n) (c(n)) to be the generating polynomials of the matchings of the rooted tree with MatulaGöbel number n that contain (do not contain) the root, with respect to the size of the matching. We have the following recurrence for the pair M(n)=[b(n),c(n)]. M(1)=[0,1]; if n=p(t) (=the tth prime), then M(n)=[xc(t),b(t)+c(t)]; if n=rs (r,s,>=2), then M(n)=[b(r)c(s)+c(r)b(s), c(r)c(s)]. Then m(n)=b(n)+c(n) is the generating polynomial of the matchings of the rooted tree with respect to the size of the matchings (called matchinggenerating polynomial). [The actual matching polynomial is obtained by the substitution x = 1/x^2, followed by multiplication by x^N(n), where N(n) is the number of vertices of the rooted tree.]


EXAMPLE

5 is in the sequence because the corresponding rooted tree is a path abcd on 4 vertices. We have 1 0matching (the empty set), 3 1matchings (ab), (bc), (cd), and 1 2matchings (ab, cd). The sequence 1,3,1 is palindromic.


MAPLE

L := 200: with(numtheory): N := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))1 end if end proc: M := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then [0, 1] elif bigomega(n) = 1 then [x*M(pi(n))[2], M(pi(n))[1]+M(pi(n))[2]] else [M(r(n))[1]*M(s(n))[2]+M(r(n))[2]*M(s(n))[1], M(r(n))[2]*M(s(n))[2]] end if end proc: m := proc (n) options operator, arrow: sort(expand(M(n)[1]+M(n)[2])) end proc: PAL := {}: for n to L do if m(n) = numer(subs(x = 1/x, m(n))) then PAL := `union`(PAL, {n}) else end if end do: PAL;


CROSSREFS

Cf. A202853.
Sequence in context: A146477 A166753 A319756 * A274911 A282536 A248719
Adjacent sequences: A202851 A202852 A202853 * A202855 A202856 A202857


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Feb 14 2012


STATUS

approved



