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A036778
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Number of labeled rooted trees on 2n+1 nodes each node having an even number of children.
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4
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1, 3, 65, 3787, 427905, 79549811, 22036379521, 8513206310715, 4374455745966593, 2885264091484122979, 2376040584184726335681, 2389484304129542889498923, 2881763610489447544905661825, 4105338427962827177938910410707, 6820519958449287654130653696838145
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OFFSET
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0,2
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.82).
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LINKS
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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
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FORMULA
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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).
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MAPLE
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[ 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
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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 *)
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PROG
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(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
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CROSSREFS
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Sequence in context: A216859 A012804 A012837 * A295169 A065400 A306410
Adjacent sequences: A036775 A036776 A036777 * A036779 A036780 A036781
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KEYWORD
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nonn,eigen
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Edited by Christian G. Bower, Jan 13 2004
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STATUS
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approved
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