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Number of trees with positive integer edge labels summing to n.
2

%I #12 Oct 06 2019 01:50:43

%S 1,1,2,4,9,21,55,146,415,1212,3653,11246,35346,112750,364714,1193202,

%T 3943557,13148575,44186841,149536376,509270554,1744342614,6005869285,

%U 20777091355,72192026878,251848377631,881865312582,3098564357293,10922162622233,38614641384893

%N Number of trees with positive integer edge labels summing to n.

%H Andrew Howroyd, <a href="/A304914/b304914.txt">Table of n, a(n) for n = 0..500</a>

%F G.f.: g(x) + (g(x^2) - g(x)^2)*x/(2*(1-x)) where g(x) is the g.f. of A052855.

%t max = 30; g[_] = 1; Do[g[x_] = Exp[Sum[(g[x^k]/(1 - x^k))*(x^k/k) + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; CoefficientList[g[x] + (g[x^2] - g[x]^2)*(x/(2*(1 - x))) + O[x]^max, x] (* _Jean-François Alcover_, May 25 2018 *)

%o (PARI) \\ here b(n) is A052855 as series

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o b(n)={my(v=[1]); for(i=2, n, v=concat([1], v + EulerT(v))); Ser(v)*(1-x)}

%o seq(n)={my(g=b(n)); Vec(g + (subst(g,x,x^2) - g^2)*x/(2*(1-x)))}

%Y Row sums of A303842.

%Y Cf. A052855.

%K nonn

%O 0,3

%A _Andrew Howroyd_, May 20 2018