A336310 Sum of path lengths over all labeled rooted unordered binary trees.
0, 0, 2, 24, 300, 4260, 69120, 1271340, 26233200, 601246800, 15171105600, 418203324000, 12509695598400, 403696590897600, 13982667790291200, 517482647165484000, 20381726051118432000, 851302665544050720000, 37587618060140244096000, 1749369290830388555328000, 85599487854917373617280000
Offset: 0
Keywords
Programs
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Mathematica
nn = 20; Range[0, nn]! CoefficientList[ Series[-(((-1 + Sqrt[1 - 2 z - z^2]) (-1 + z + Sqrt[1 - 2 z - z^2]))/(z (-1 + 2 z + z^2))), {z, 0, nn}], z] -
PARI
my(z='z+O('z^25)); concat([0,0], Vec(serlaplace(((1 -sqrt(1 -2*z -z^2))*(1 -z -sqrt(1 -2*z -z^2)))/(z*(1 -2*z -z^2))))) \\ Joerg Arndt, Jul 18 2020
Formula
E.g.f.: ((1 -sqrt(1 -2*z -z^2))*(1 -z -sqrt(1 -2*z -z^2)))/(z*(1 -2*z -z^2)).
a(n) = Sum_{k} A336309(n,k)*k, for n>=1.
a(n) ~ n!/2 * (sqrt(2) + 1)^(n+1) * (1 - sqrt((10-sqrt(2))/(Pi*n))). - Vaclav Kotesovec, Jul 17 2020