Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #39 Jun 23 2019 22:14:53
%S 1,5,2520,9909900,150089940000,6217438242015000,574985352122181000000,
%T 103753754577643425255000000,33189544956070738228953960000000,
%U 17517292900368819935211385551000000000,14427024664929016470240101675459976000000000
%N Number of labeled rooted trees with a degree constraint: ((4*n)!/(24^n)) * binomial(4*n+1, n).
%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; see Eq. (13) on p. 4 (with r = 4).
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F From _Petros Hadjicostas_, Jun 08 2019: (Start)
%F Recurrence (with no interpolated zeros): -8 * (4*n + 1) * (4*n + 3)^2 * (2*n + 1)^2 * (4*n + 5) * a(n) + (81*n^2 + 162*n + 72) * a(n + 1) = 0 for n >= 0 with a(0) = 1.
%F E.g.f. (with interpolated zeros): Let G(x) = Sum_{n >= 0} a(n)*x^(4*n + 1)/(4*n + 1)!. Then the e.g.f. satisfies G(x) = x * (1 + G(x)^4/4!).
%F (End)
%t Table[(4n)!/24^n Binomial[4n+1,n],{n,0,10}] (* _Harvey P. Dale_, Aug 10 2011 *)
%Y Cf. A036770, A036771, A036773.
%K nonn
%O 0,2
%A _N. J. A. Sloane_.