OFFSET
0,5
COMMENTS
T(n, k) where n counts the vertices and 0 <= k <= n counts the labels. - Sean A. Irvine, Mar 22 2018
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.
LINKS
Sean A. Irvine, Table of n, a(n) for n = 0..527
J. Riordan, The numbers of labeled colored and chromatic trees, Acta Mathematica 97 (1957) 211.
FORMULA
E.g.f.: r(x,y) = T(n,k) * y^k * x^n / k! satisfies r(x,y) * exp(r(x)) = (1+y) * r(x) * exp(r(x,y)) where r(x) is the o.g.f. for A000081. - Sean A. Irvine, Mar 22 2018
EXAMPLE
Triangle begins with T(0,0):
n\k 0 1 2 3 4 5 6
0 1
1 1 1
2 1 2 2
3 2 5 9 9
4 4 13 34 64 64
5 9 35 119 326 625 625
6 20 95 401 1433 4016 7776 7776
MATHEMATICA
m = 9; r[_] = 0;
Do[r[x_] = x Exp[Sum[r[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, m}];
r[x_, y_] = -ProductLog[(-E^(-r[x])) r[x] - (r[x] y)/E^r[x]];
(CoefficientList[#, y] Range[0, Exponent[#, y]]!)& /@ CoefficientList[r[x, y] + O[x]^m, x] /. {} -> {1} // Flatten // Quiet (* Jean-François Alcover, Oct 23 2019 *)
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, Mar 22 2018
Name edited by Andrew Howroyd, Mar 23 2023
STATUS
approved