OFFSET
0,8
COMMENTS
Not all k colors need to be used. The total number of nodes will be 2n-1.
See table 4.1 in the Johnson reference.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. South Carolina, 2012.
FORMULA
G.f. of column k: 1 + R(x) + (R(x^3) - R(x)^3)/3 where R(x) is the g.f. of column k of A319539.
EXAMPLE
Array begins:
======================================================
n\k| 0 1 2 3 4 5 6 7
---+--------------------------------------------------
0 | 1 1 1 1 1 1 1 1 ...
1 | 0 1 2 3 4 5 6 7 ...
2 | 0 1 3 6 10 15 21 28 ...
3 | 0 1 4 10 20 35 56 84 ...
4 | 0 1 6 21 55 120 231 406 ...
5 | 0 1 12 63 220 600 1386 2842 ...
6 | 0 2 31 227 1040 3530 9772 23366 ...
7 | 0 2 78 891 5480 23250 77112 214718 ...
8 | 0 4 234 3876 31420 165510 655599 2122099 ...
9 | 0 6 722 17790 190360 1243825 5878446 22102577 ...
...
PROG
(PARI) \\ here U(n, k) gives column k as a vector.
R(n, k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}
U(n, k)={my(g=x*Ser(R(n, k))); Vec(1 + g + (subst(g + O(x*x^(n\3)), x, x^3) - g^3)/3)}
{my(T=Mat(vector(8, k, U(8, k-1)~))); for(n=1, #T~, print(T[n, ]))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Dec 14 2020
STATUS
approved