

A238414


Triangle read by rows: T(n,k) is the number of trees with n vertices having maximum vertex degree k (n>=3, 2<=k<=n1).


1



1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 1, 10, 7, 3, 1, 1, 1, 17, 17, 7, 3, 1, 1, 1, 36, 38, 19, 7, 3, 1, 1, 1, 65, 93, 45, 19, 7, 3, 1, 1, 1, 134, 220, 118, 47, 19, 7, 3, 1, 1, 1, 264, 537, 296, 125, 47, 19, 7, 3, 1, 1, 1, 551, 1306, 775, 321, 127, 47, 19, 7, 3, 1, 1
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OFFSET

3,8


COMMENTS

Sum of entries in row n is A000055(n) (= number of trees with n vertices).


LINKS

Table of n, a(n) for n=3..80.
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)


FORMULA

The author knows of no formula for T(n,k). The entries have been obtained in the following manner, explained for row n = 7. In A235111 we find that the 11 (= A000055(7)) trees with 7 vertices have Mindices 25, 27, 30, 35, 36, 40, 42, 48, 49, 56, and 64 (the Mindex of a tree t is the smallest of the Matula numbers of the rooted trees isomorphic, as a tree, to t). Making use of the formula in A196046, from these Matula numbers one obtains the maximum vertex degrees: 2, 3, 3, 3, 4, 4, 3, 5, 3, 4, 6; the frequencies of 2,3,4,5,6 are 1, 5, 3, 1, 1, respectively. See the Maple program.


EXAMPLE

Row n=4 is T(4,2)=1,T(4,3)=1; indeed, the maximum vertex degree in the path P[4] is 2, while in the star S[4] it is 3.
Triangle starts:
1;
1, 1;
1, 1, 1;
1, 3, 1, 1;
1, 5, 3, 1, 1;
1, 10, 7, 3, 1, 1;
1, 17, 17, 7, 3, 1, 1;


MAPLE

MI := [25, 27, 30, 35, 36, 40, 42, 48, 49, 56, 64]: with(numtheory): a := 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 max(a(pi(n)), 1+bigomega(pi(n))) else max(a(r(n)), a(s(n)), bigomega(r(n))+bigomega(s(n))) end if end proc: g := add(x^a(MI[j]), j = 1 .. nops(MI)): seq(coeff(g, x, q), q = 2 .. 6);


CROSSREFS

Cf. A235111, A196046.
Sequence in context: A255811 A131268 A109221 * A046643 A254101 A277604
Adjacent sequences: A238411 A238412 A238413 * A238415 A238416 A238417


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Mar 05 2014


STATUS

approved



