

A198333


Irregular triangle read by rows: row n is the pruning partition of the rooted tree with MatulaGoebel number n.


2



1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 1, 4, 2, 3, 1, 2, 3, 1, 4, 1, 2, 3, 1, 2, 3, 2, 2, 2, 1, 2, 3, 2, 4, 1, 2, 2, 2, 1, 2, 3, 1, 3, 3, 2, 4, 1, 2, 3, 2, 2, 3, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 3, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

The pruning partition of a tree is the reverse sequence of the number of vertices of degree 1, deleted at the successive prunings. By pruning we mean the deletion of vertices of degree 1 and of their incident edges. See the Balaban reference (p. 360) and/or the TodeschiniConsonni reference (p. 42).
The MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


REFERENCES

A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355375, 1979.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, WileyVCH, 2000.


LINKS

Table of n, a(n) for n=1..86.
F. GĂ¶bel, 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


FORMULA

A198329(n) is the MatulaGoebel number of the rooted tree obtained by removing from the rooted tree with MatulaGoebel number n the vertices of degree one, together with their incident edges. Repeated application of this yields the MatulaGoebel numbers of the trees obtained by successive prunings. Finding the number of vertices of these trees and taking differences lead to the pruning partition (see the Maple program and the explanation given there).


EXAMPLE

Row 7 is 1,3 because the rooted tree with MatulaGoebel number 7 is Y, having 3 vertices of degree 1 and after the first pruning we obtain the 1vertex tree.
The triangle starts:  Squared  Sum of squares (= A198334(n)).
1; 1; 1
2; 4; 4
1,2; 1,4; 5
1,2; 1,4; 5
2,2; 4,4; 8
2,2; 4,4; 8
1,3; 1,9; 10
1,3; 1,9; 10
1,2,2; 1,4,4; 9
 edited by Antti Karttunen, Mar 07 2017


MAPLE

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: a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: MS := proc (m) local A, i: A[m, 1] := m: for i from 2 while 1 < A[m, i1] do A[m, i] := a(A[m, i1]) end do: if A[m, i2] = 2 then [seq(A[m, j], j = 1 .. i2)] else [seq(A[m, j], j = 1 .. i1)] end if end proc: PP := proc (n) local NVP, q: q := nops(MS(n)): NVP := map(N, MS(n)): NVP[q], seq(NVP[qj]NVP[qj+1], j = 1 .. nops(NVP)1) end proc: for n to 21 do PP(n) end do; # for the rooted tree with MatulaGoebel number n, N(n)=A061775(n) is the number of vertices, a(n) (=A198329(n)) is the MatulaGoebel number of the tree obtained after one pruning, MS(n) is the sequence of MatulaGoebel numbers of the trees obtained after 0, 1, 2, ... prunings, PP(n) is the pruning partition, i.e. the number of vertices of degree 1 deleted at the successive prunings, given in reverse order.


CROSSREFS

Cf. A061775, A198329, A198334.
Sequence in context: A260341 A109969 A085035 * A191591 A083023 A084359
Adjacent sequences: A198330 A198331 A198332 * A198334 A198335 A198336


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Nov 25 2011


STATUS

approved



