A036778 Number of labeled rooted trees on 2n+1 nodes each node having an even number of children.
1, 3, 65, 3787, 427905, 79549811, 22036379521, 8513206310715, 4374455745966593, 2885264091484122979, 2376040584184726335681, 2389484304129542889498923, 2881763610489447544905661825, 4105338427962827177938910410707, 6820519958449287654130653696838145
Offset: 0
References
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.82).
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..210
- Yiyang Jia and Jacobus J. M. Verbaarschot, Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs, arXiv:1806.03271 [hep-th], 2018.
- Yiyang Jia and Jacobus J. M. Verbaarschot, Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs, J. High Energ. Phys. (2018) 2018: 31.
- L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (16).
- Index entries for sequences related to rooted trees
Programs
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Maple
[ seq((1/2^(2*n+1))*add( binomial(2*n+1,j)*(2*j-(2*n+1))^(2*n),j=0..(2*n+1)), n=1..30) ];
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Mathematica
Table[1/2^(2n+1) Sum[Binomial[2n+1,k](2k-2n-1)^(2n),{k,0,2n+1}],{n,0,20}] (* Harvey P. Dale, Mar 06 2012 *) -
PARI
a(n)=local(X); if(n<0,0,X=x+O(x^(2*n+1));(2*n+1)!*polcoeff(serreverse(x/cosh(x)),2*n+1)) \\ Paul D. Hanna, Oct 15 2003
Formula
G.f.: REVERT(x/cosh(x)) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!. - Paul D. Hanna, Oct 15 2003
a(n) = (1/2^(2*n+1)) * Sum_{k=0..2*n+1} binomial(2*n+1, k)*(2*k-2*n-1)^(2*n).
Extensions
Edited by Christian G. Bower, Jan 13 2004