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

%I A055544 #25 Feb 16 2025 08:32:43
%S A055544 1,2,6,16,45,120,336,920,2574,7190,20262,57192,162318,461622,1317165,
%T A055544 3766096,10792399,30980862,89084844,256524560,739658472,2135213982,
%U A055544 6170505665,17849399616,51679366125,149750549260,434260826883,1260197855532,3659410074932
%N A055544 Total number of nodes in all rooted trees with n nodes.
%H A055544 Alois P. Heinz, <a href="/A055544/b055544.txt">Table of n, a(n) for n = 1..1000</a>
%H A055544 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphVertex.html">Graph Vertex.</a>
%H A055544 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphEdge.html">Graph Edge</a>
%H A055544 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A055544 a(n) = n*A000081(n).
%F A055544 G.f.: x * (d/dx) x * Product_{k>=1} 1/(1 - x^k)^(a(k)/k). - _Ilya Gutkovskiy_, May 26 2019
%p A055544 with(numtheory):
%p A055544 b:= proc(n) option remember; local d, j; `if`(n<2, n,
%p A055544       (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1))
%p A055544     end:
%p A055544 a:= n-> n*b(n):
%p A055544 seq(a(n), n=1..30);  # _Alois P. Heinz_, May 16 2013
%t A055544 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_ *)
%Y A055544 Cf. A000081, A000169, A055543, A055544.
%K A055544 nonn
%O A055544 1,2
%A A055544 _Eric W. Weisstein_