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A036778 Number of labeled rooted trees on 2n+1 nodes each node having an even number of children. 5

%I #34 Jun 09 2019 19:06:50

%S 1,3,65,3787,427905,79549811,22036379521,8513206310715,

%T 4374455745966593,2885264091484122979,2376040584184726335681,

%U 2389484304129542889498923,2881763610489447544905661825,4105338427962827177938910410707,6820519958449287654130653696838145

%N Number of labeled rooted trees on 2n+1 nodes each node having an even number of children.

%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.82).

%H Michael De Vlieger, <a href="/A036778/b036778.txt">Table of n, a(n) for n = 0..210</a>

%H Yiyang Jia and Jacobus J. M. Verbaarschot, <a href="https://arxiv.org/abs/1806.03271">Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs</a>, arXiv:1806.03271 [hep-th], 2018.

%H Yiyang Jia and Jacobus J. M. Verbaarschot, <a href="https://doi.org/10.1007/JHEP11(2018)031">Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs</a>, J. High Energ. Phys. (2018) 2018: 31.

%H L. Takacs, <a href="http://www.appliedprobability.org/data/files/TMS%20articles/18_1_1.pdf">Enumeration of rooted trees and forests</a>, Math. Scientist 18 (1993), 1-10, esp. Eq. (16).

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F G.f.: REVERT(x/cosh(x)) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!. - _Paul D. Hanna_, Oct 15 2003

%F 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).

%p [ 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) ];

%t 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 *)

%o (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

%K nonn,eigen

%O 0,2

%A _N. J. A. Sloane_

%E Edited by _Christian G. Bower_, Jan 13 2004

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