A036770 - OEIS

%I #69 Sep 08 2022 08:44:52

%S 1,3,60,3150,317520,52390800,12843230400,4382752374000,

%T 1986847742880000,1155153277710432000,838011196011749760000,

%U 742058914068404412480000,787724078011075453248000000,987468397792455300321600000000,1443283810213452666950050560000000

%N Number of labeled rooted trees with a degree constraint: (2*n)!/(2^n) * C(2*n+1, n).

%C a(n) is the number of rooted labeled strictly binary trees (each vertex has exactly two children or none) on 2*n + 1 vertices. - _Geoffrey Critzer_, Nov 13 2011

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=46">Encyclopedia of Combinatorial Structures 46</a>.

%H Lajos 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; see Eqs. (12) and (13) on p. 4.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StronglyBinaryTree.html">Strongly Binary Tree</a>.

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

%F Recurrence (with interpolated zeros): Define (b(n): n >= 0) by b(2*m+1) = a(m) and b(2*m) = 0 for m >= 0. Then the sequence (b(n): n >= 0) satisfies the recurrence (-2*n^3 - 6*n^2 - 4*n)*b(n) + (n + 3)*b(n+2) = 0 for n >= 0 with b(0) = 0 and b(1) = 1. [Corrected by _Petros Hadjicostas_, Jun 07 2019]

%F E.g.f. with interpolated zeros: G(x) = Sum_{n >= 0} b(n)*x^n/n! = Sum_{m >= 0} a(m)*x^(2*m + 1)/(2*m + 1)! = 1/x * (1 - (1 - 2*x^2)^(1/2)) for 0 < |x| < 1/sqrt(2). [Edited by _Petros Hadjicostas_, Jun 07 2019]

%F E.g.f. with interpolated zeros satisfies G(x)= x*(1 + G(x)^2/2). - _Geoffrey Critzer_, Nov 13 2011

%F D-finite with recurrence (with no interpolated zeros): -2*a(n)*(n + 1)*(2*n + 1)*(2*n + 3) + (n + 2)*a(n+1) = 0 with a(0) = 1. - _Petros Hadjicostas_, Jun 08 2019

%F G.f.: 4F1(1,1,1/2,3/2;2;8*x). - _R. J. Mathar_, Jan 28 2020

%p spec := [S,{S=Union(Z,Prod(Z,Set(S,card=2)))},labeled]: seq(combstruct[count](spec,size=n)

%t Range[0, 19]! CoefficientList[Series[(1 - (1 - 2 x^2)^(1/2))/x, {x, 0, 20}], x] (* _Geoffrey Critzer_, Nov 13 2011 *)

%o (Magma) [Factorial(2*n)/(2^n) * Binomial(2*n+1, n): n in [0..15]]; // _Vincenzo Librandi_, Jan 29 2020

%Y Cf. A001190, A036771, A036772, A036773, A052510.

%K nonn

%O 0,2

%A _N. J. A. Sloane_