A336255 Irregular triangular array read by rows. T(n,k) is the number of rooted labeled trees on n nodes with path length exactly k, n>=1, 0<=k<=C(n,2).
1, 0, 2, 0, 0, 3, 6, 0, 0, 0, 4, 24, 12, 24, 0, 0, 0, 0, 5, 60, 120, 140, 120, 60, 120, 0, 0, 0, 0, 0, 6, 120, 540, 840, 1470, 720, 1440, 840, 720, 360, 720, 0, 0, 0, 0, 0, 0, 7, 210, 1680, 4620, 9240, 11382, 13440, 14700, 10920, 12810, 10080, 10080, 5880, 5040, 2520, 5040
Offset: 1
Examples
1, 0, 2, 0, 0, 3, 6, 0, 0, 0, 4, 24, 12, 24, 0, 0, 0, 0, 5, 60, 120, 140, 120, 60, 120
Links
- Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 185.
Programs
-
Mathematica
nn = 7; f[z_, u_] := Sum[Sum[a[n, k] u^k z^n/n!, {k, 0, Binomial[n, 2]}], {n, 1, nn}]; sol = SolveAlways[ Series[0 == f[z, u] - z Exp[f[u z, u]] , {z, 0, nn}], {z, u}];Level[Table[Table[a[n, k], {k, 0, Binomial[n, 2]}], {n, 1, nn}] /. sol, {2}] // Grid
Formula
E.g.f. satisfies A(x,y) = x*exp(A(y*x,y)).
Sum_{k=n-1..C(n,2)} T(n,k)*k = A001864(n).
Comments