

A257537


Number of subtrees with at least one edge of the rooted tree with MatulaGoebel number n.


0



0, 1, 3, 3, 6, 6, 7, 7, 10, 10, 10, 12, 12, 12, 15, 15, 12, 19, 15, 18, 18, 15, 19, 24, 21, 19, 29, 22, 18, 27, 15, 31, 21, 18, 25, 37, 24, 24, 27, 34, 19, 33, 22, 25, 40, 29, 27, 48, 30, 37, 25, 33, 31, 56, 28, 42, 34, 27, 18, 51, 37, 21, 49, 63, 36, 36, 24, 30, 40, 45, 34, 73, 33, 37, 54, 42, 33, 48, 25
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OFFSET

1,3


COMMENTS

The Matula 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 Matula 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 numbers of the m branches of T.
a(n) = A184161(n)  A061775(n).
The subtree polynomial of a tree T is the bivariate generating polynomial of the subtrees of T with at least one edge with respect to the number of edges (marked by x) and number of pendant vertices (marked by y). See the Martin et. al, reference. For example, the subtree polynomial of the tree \/ is 2xy^2 + x^2 y^2.
If G(n;x,y) is the subtree polynomial of the rooted tree with Matula number n, then, obviously, a(n) = G(n;1,1). The Maple program uses this circuitous way for finding a(n). With the given Maple program, the command G(n) yields the subtree polynomial of the rooted tree having Matula number n (this is my "secret" reason to include this sequence).


LINKS

Table of n, a(n) for n=1..79.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 23142322.
F. Goebel, 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. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


EXAMPLE

a((4)=3 because the rooted tree with Matula number 4 is \/ with subtrees \ , / , and \/ .


MAPLE

with(numtheory): g := 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 elif bigomega(n) = 1 then expand(x*y^2+x*g(pi(n))+x*y*h(pi(n))) else expand(g(r(n))+g(s(n))) end if end proc: h := 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 elif bigomega(n) = 1 then 0 else expand(h(r(n))+h(s(n))+g(r(n))*g(s(n))/y^2+g(r(n))*h(s(n))/y+h(r(n))*g(s(n))/y+h(r(n))*h(s(n))) end if end proc: G := 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 elif bigomega(n) = 1 then g(n)+G(pi(n)) else expand(G(r(n))+G(s(n))+h(n)h(r(n))h(s(n))) end if end proc: seq(subs({x = 1, y = 1}, G(i)), i = 1 .. 150);


CROSSREFS

Cf. A184161, A061775
Sequence in context: A127739 A175394 A070318 * A219883 A219381 A219852
Adjacent sequences: A257534 A257535 A257536 * A257538 A257539 A257540


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Apr 28 2015


STATUS

approved



