A055543 Total number of nodes in all trees with n nodes.

1, 2, 3, 8, 15, 36, 77, 184, 423, 1060, 2585, 6612, 16913, 44226, 116115, 309120, 826693, 2229606, 6041145, 16461300, 45034605, 123722632, 341045702, 943197528, 2615922250, 7274629700, 20278767420, 56656404896, 158617430965, 444926154060, 1250255699930

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Graph Vertex.
• Index entries for sequences related to trees

Crossrefs

Cf. A000055, A000169, A055542, A055544.

Programs

Mathematica

nn = 25; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol = SolveAlways[0 == Series[
    f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x];
r[x_] := Sum[a[n] x^n, {n, 0, nn}] /. sol; CoefficientList[Series[x D[r[x] - 1/2 (r[x]^2 - r[x^2]), x], {x, 0, nn}], x] (* Geoffrey Critzer, Jul 06 2020 *)

Formula

a(n) = n*A000055(n).

O.g.f.: x d/dx A(x) where A(x) is the o.g.f. for A000055. - Geoffrey Critzer, Jul 06 2020

Extensions

More terms, formula from Christian G. Bower, Jun 12 2000