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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).

%I A336255 #12 Jul 18 2020 03:06:33
%S A336255 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,
%T A336255 0,0,6,120,540,840,1470,720,1440,840,720,360,720,0,0,0,0,0,0,7,210,
%U A336255 1680,4620,9240,11382,13440,14700,10920,12810,10080,10080,5880,5040,2520,5040
%N 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).
%C A336255 The path length of a tree is the distance from the root to a node summed over all nodes in the tree.
%H A336255 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 185.
%F A336255 E.g.f. satisfies A(x,y) = x*exp(A(y*x,y)).
%F A336255 Sum_{k=n-1..C(n,2)} T(n,k)*k = A001864(n).
%e A336255 1,
%e A336255 0, 2,
%e A336255 0, 0, 3, 6,
%e A336255 0, 0, 0, 4, 24, 12, 24,
%e A336255 0, 0, 0, 0, 5,  60, 120, 140, 120, 60, 120
%t A336255 nn = 7; f[z_, u_] := Sum[Sum[a[n, k] u^k z^n/n!, {k, 0, Binomial[n, 2]}], {n, 1,
%t A336255    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}] /.
%t A336255    sol, {2}] // Grid
%Y A336255 Row sums give A000169.
%Y A336255 Cf. A001864.
%K A336255 nonn,tabf
%O A336255 1,3
%A A336255 _Geoffrey Critzer_, Jul 14 2020