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A336309 Triangular array read by rows. T(n,k) is the number of labeled rooted unordered binary trees (as in A036774) with path length exactly k, n >= 1, 0 <= k <= C(n,2).

%I A336309 #15 Jul 18 2020 03:06:46
%S A336309 1,0,2,0,0,3,6,0,0,0,0,24,12,24,0,0,0,0,0,0,120,120,120,60,120,0,0,0,
%T A336309 0,0,0,0,0,360,1440,360,1440,720,720,360,720,0,0,0,0,0,0,0,0,0,0,630,
%U A336309 7560,10080,10080,7560,12600,7560,10080,5040,5040,2520,5040
%N A336309 Triangular array read by rows. T(n,k) is the number of labeled rooted unordered binary trees (as in A036774) with path length exactly k, n >= 1, 0 <= k <= C(n,2).
%C A336309 The path length of a tree is the distance from the root to a node summed over all nodes in the tree.
%F A336309 E.g.f. satisfies A(x,y) = x + x*A(y*x,y) + x*A(y*x,y)^2/2.
%e A336309   1,
%e A336309   0, 2,
%e A336309   0, 0, 3, 6,
%e A336309   0, 0, 0, 0, 24, 12,  24,
%e A336309   0, 0, 0, 0,  0,  0, 120, 120, 120,   60,   120,
%e A336309   0, 0, 0, 0,  0,  0,   0,   0, 360, 1440,   360, 1440, 720, 720, 360, 720
%t A336309 nn = 6; f[z_, u_] := Sum[Sum[a[n, k] u^k z^n/n!, {k, 0, Binomial[n, 2]}], {n, 1,
%t A336309    nn}]; sol =SolveAlways[Series[0 == f[z, u] - z (1 + f[u z, u] + f[u z, u]^2/2!), {z, 0, nn}], {z, u}];Level[Table[Table[a[n, k], {k, 0, Binomial[n, 2]}], {n, 1, nn}] /. sol, {2}] // Grid
%Y A336309 Cf. A036774.
%K A336309 nonn,tabf
%O A336309 1,3
%A A336309 _Geoffrey Critzer_, Jul 17 2020