A238414 Triangle read by rows: T(n,k) is the number of trees with n vertices having maximum vertex degree k (n>=1, 0<=k<=n-1).

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 3, 1, 1, 0, 0, 1, 5, 3, 1, 1, 0, 0, 1, 10, 7, 3, 1, 1, 0, 0, 1, 17, 17, 7, 3, 1, 1, 0, 0, 1, 36, 38, 19, 7, 3, 1, 1, 0, 0, 1, 65, 93, 45, 19, 7, 3, 1, 1, 0, 0, 1, 134, 220, 118, 47, 19, 7, 3, 1, 1, 0, 0, 1, 264, 537, 296, 125, 47, 19, 7, 3, 1, 1

Offset: 1

Author

Emeric Deutsch, Mar 05 2014

Keywords

• nonn
• tabl

Comments

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

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 M-indices 25, 27, 30, 35, 36, 40, 42, 48, 49, 56, and 64 (the M-index 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.

This sequence may be derived from A144528 which can be efficiently computed in the same manner as A000055. - Andrew Howroyd, Dec 17 2020

Examples

			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;
  0, 1;
  0, 0, 1;
  0, 0, 1,  1;
  0, 0, 1,  1,  1;
  0, 0, 1,  3,  1, 1;
  0, 0, 1,  5,  3, 1, 1;
  0, 0, 1, 10,  7, 3, 1, 1;
  0, 0, 1, 17, 17, 7, 3, 1, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
• 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.)
• Eric Weisstein's World of Mathematics, Maximum Vertex Degree

Crossrefs

Row sums are A000055.

Cf. A144528, A196046, A235111, A332760 (connected graphs), A339788 (forests).

Programs

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);

PARI

\\ Here V(n, k) gives column k of A144528.
MSet(p,k)={my(n=serprec(p,x)-1); if(min(k,n)<1, 1 + O(x*x^n), polcoef(exp( sum(i=1, min(k,n), (y^i + O(y*y^k))*subst(p + O(x*x^(n\i)), x, x^i)/i ))/(1-y + O(y*y^k)), k, y))}
V(n,k)={my(g=1+O(x)); for(n=2, n, g=x*MSet(g, k-1)); Vec(1 + x*MSet(g, k) + (subst(g, x, x^2) - g^2)/2)}
M(n, m=n)={my(v=vector(m, k, V(n,k-1)[2..1+n]~)); Mat(vector(m, k, v[k]-if(k>1, v[k-1])))}
{ my(T=M(12)); for(n=1, #T~, print(T[n, 1..n])) } \\ Andrew Howroyd, Dec 18 2020

Formula

T(n,k) = A144528(n,k) - A144528(n, k-1) for k > 0. - Andrew Howroyd, Dec 17 2020

Extensions

Columns k=0..1 inserted by Andrew Howroyd, Dec 18 2020