A055544 Total number of nodes in all rooted trees with n nodes.

1, 2, 6, 16, 45, 120, 336, 920, 2574, 7190, 20262, 57192, 162318, 461622, 1317165, 3766096, 10792399, 30980862, 89084844, 256524560, 739658472, 2135213982, 6170505665, 17849399616, 51679366125, 149750549260, 434260826883, 1260197855532, 3659410074932

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Graph Vertex.
• Eric Weisstein's World of Mathematics, Graph Edge
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A000081, A000169, A055543, A055544.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<2, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1))
    end:
a:= n-> n*b(n):
seq(a(n), n=1..30);  # Alois P. Heinz, May 16 2013

Mathematica

b[n_] := b[n] = If[n<2, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}])/(n-1)]; a[n_] := n*b[n]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 25 2017, after Alois P. Heinz *)

Formula

a(n) = n*A000081(n).

G.f.: x * (d/dx) x * Product_{k>=1} 1/(1 - x^k)^(a(k)/k). - Ilya Gutkovskiy, May 26 2019