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Total number of nodes in all trees with n nodes.
4

%I #17 Jul 06 2020 17:04:39

%S 1,2,3,8,15,36,77,184,423,1060,2585,6612,16913,44226,116115,309120,

%T 826693,2229606,6041145,16461300,45034605,123722632,341045702,

%U 943197528,2615922250,7274629700,20278767420,56656404896,158617430965,444926154060,1250255699930

%N Total number of nodes in all trees with n nodes.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphVertex.html">Graph Vertex.</a>

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

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

%F O.g.f.: x d/dx A(x) where A(x) is the o.g.f. for A000055. - _Geoffrey Critzer_, Jul 06 2020

%t nn = 25; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol = SolveAlways[0 == Series[

%t f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x];

%t 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 *)

%Y Cf. A000055, A000169, A055542, A055544.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_

%E More terms, formula from _Christian G. Bower_, Jun 12 2000