|
%I #38 Sep 10 2022 01:51:49
%S 1,1,2,5,13,39,121,388,1277,4288,14630,50575,176762,623563,2217379,
%T 7939821,28603591,103600632,377033451,1378023887,5056021292,
%U 18615654196,68758804039,254706453524,946038872000,3522439937992,13145057553230,49157901220299
%N Ordered trees with branches of length at most 3.
%H G. C. Greubel, <a href="/A239106/b239106.txt">Table of n, a(n) for n = 0..1000</a>
%H Beáta Bényi, Toufik Mansour, and José L. Ramírez, <a href="https://ajc.maths.uq.edu.au/pdf/84/ajc_v84_p325.pdf">Set partitions and non-crossing partitions with l-neighbors and l-isolated elements</a>, Australasian J. Comb. (2022) Vol. 84, No. 2, 325-340.
%H Emeric Deutsch, <a href="http://dx.doi.org/10.1016/j.disc.2003.10.021">Ordered trees with prescribed root degrees, node degrees and branch lengths</a>, Discrete Math., 282, 2004, 89-94. See p. 93.
%F a(n) = Sum_{k= ceiling(n/3)..n} trinomial(k,3;n-k) A001006(k-1).
%F G.f.: (1+x+x^2+x^3-sqrt(-3*x^6-6*x^5-9*x^4-8*x^3-5*x^2-2*x+1))/ (2*x^3+2*x^2+2*x). - _Vladimir Kruchinin_, Mar 03 2016
%F Recurrence: (n+1)*a(n) = (n-2)*a(n-1) + 6*(n-2)*a(n-2) + 3*(5*n - 13)*a(n-3) + (22*n - 83)*a(n-4) + (23*n - 112)*a(n-5) + 18*(n-6)*a(n-6) + 9*(n-7)*a(n-7) + 3*(n-8)*a(n-8). - _Vaclav Kotesovec_, Mar 03 2016
%F a(n) ~ sqrt(3*(3 - 1/(3-2*sqrt(2))^(1/3) - (3-2*sqrt(2))^(1/3))/2) * ((108-54*sqrt(2))^(1/3)/3 + (4+2*sqrt(2))^(1/3) + 1)^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 03 2016
%p beta := proc(n,k)
%p coeftayl((x+x^2+x^3)^k,x=0,n) ;
%p end proc:
%p A239106 := proc(n)
%p add( A001006(k-1)*beta(n,k),k=ceil(n/3)..n) ;
%p end proc: # _R. J. Mathar_, Apr 02 2014
%t CoefficientList[Series[(1 + x + x^2 + x^3 - Sqrt[-3*x^6 - 6*x^5 - 9*x^4 - 8*x^3 - 5*x^2 - 2*x + 1])/(2*x^3 + 2*x^2 + 2*x), {x, 0, 50}], x] (* _G. C. Greubel_, Apr 05 2017 *)
%o (PARI) x='x+O('x^50); Vec((1+x+x^2+x^3-sqrt(-3*x^6-6*x^5-9*x^4-8*x^3-5*x^2-2*x+1))/ (2*x^3+2*x^2+2*x)) \\ _G. C. Greubel_, Apr 05 2017
%Y Cf. A001006.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Mar 26 2014
|
