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

Author

Geoffrey Critzer, Jul 17 2020

Keywords

• nonn

Crossrefs

Cf. A336309, A036774 (row sums).

Programs

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