OFFSET
1,3
COMMENTS
The path length of a tree is the distance from the root to a node summed over all nodes in the tree.
FORMULA
E.g.f. satisfies A(x,y) = x + x*A(y*x,y) + x*A(y*x,y)^2/2.
EXAMPLE
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, 0, 0, 0, 0, 0, 360, 1440, 360, 1440, 720, 720, 360, 720
MATHEMATICA
nn = 6; 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 (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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Jul 17 2020
STATUS
approved