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A139263 Number of two element anti-chains in Riordan trees on n edges (also called non-redundant trees, i.e., ordered trees where no vertex has out-degree equal to 1).

%I A139263 #31 Sep 27 2024 20:54:55
%S A139263 0,0,1,3,14,48,172,580,1941,6373,20725,66763,213575,679141,2148948,
%T A139263 6771068,21257741,66529077,207639925,646480555,2008458669,6227766899,
%U A139263 19277394308,59577651108,183865477474,566700165898,1744578701517,5364804428455,16480883532586,50582859417868,155114365434224
%N A139263 Number of two element anti-chains in Riordan trees on n edges (also called non-redundant trees, i.e., ordered trees where no vertex has out-degree equal to 1).
%H A139263 Alois P. Heinz, <a href="/A139263/b139263.txt">Table of n, a(n) for n = 0..2093</a>
%H A139263 Lifoma Salaam, <a href="https://search.proquest.com/docview/193997569">Combinatorial statistics on phylogenetic trees</a>, Ph.D. Dissertation, Howard University, Washington D.C., 2008; see p. 40.
%F A139263 G.f.: A(x) = x^2 * (x*M(x) + 1)^3 * (d(x*R(x))/dx)^3/R(x), where M is the generating function for the Motzkin numbers and R is the generating function for the Riordan numbers.
%F A139263 From _Petros Hadjicostas_, Jun 02 2020: (Start)
%F A139263 Here R(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*x*(1-x)) = g.f. of A005043 and M(x) = (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2) = g.f. of A001006.
%F A139263 If we let s(x) = sqrt(1 - 2*x - 3*x^2), then A(x) = x^2*((1 + x - s(x))/(2*x*(1 + x)))^2/s(x)^3 (see p. 40 in Salaam (2008)).
%F A139263 Alternatively, we may write A(x) = x^2 * R(x)^2 * B(x), where B(x) = g.f. of (A102839(n+1): n >= 0). (End)
%p A139263 a:= proc(n) option remember; `if`(n<4, [0$2, 1, 3][n+1],
%p A139263       ((4*n-3)*(n-2)*a(n-1)+(2*n+9)*(n-2)*a(n-2)-3*
%p A139263        (4*n-9)*n*a(n-3)-9*(n-1)*n*a(n-4))/(n*(n-2)))
%p A139263     end:
%p A139263 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 02 2020
%t A139263 CoefficientList[Series[(1 -x -Sqrt[1-2*x-3*x^2])*Sqrt[1-2*x-3*x^2]/(2*(1+x)*(1 - 2*x-3*x^2)^2), {x, 0, 35}], x] (* _G. C. Greubel_, Jun 02 2020 *)
%o A139263 (Sage) r(x)=(1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x))
%o A139263 m(x)=(1-x-sqrt(1-2*x-3*x^2))/(2*x^2)
%o A139263 g(x)=derivative(x*r(x),x)
%o A139263 a(x)=x^2*(x*m(x)+1)^3*g(x)^3/r(x)
%o A139263 taylor(a(x),x,0,30).list() # _Petros Hadjicostas_, Jun 02 2020
%o A139263 (PARI) default(seriesprecision, 50)
%o A139263 f(x) = 1/sqrt(1-2*x-3*x^2);
%o A139263 r(x) = (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x));
%o A139263 a(n) = polcoef(x^2*r(x)^2*f(x)^3, n, x);
%o A139263 for(n=0, 30, print1(a(n), ",")) \\ _Petros Hadjicostas_, Jun 02 2020
%o A139263 (Magma) R<x>:=PowerSeriesRing(Rationals(), 35); [0,0] cat Coefficients(R!( (1 -x -Sqrt(1-2*x-3*x^2))*Sqrt(1-2*x-3*x^2)/(2*(1+x)*(1-2*x-3*x^2)^2) )); // _G. C. Greubel_, Jun 02 2020
%Y A139263 Cf. A001006, A005043, A102839, A139262, A178834.
%K A139263 nonn
%O A139263 0,4
%A A139263 _Lifoma Salaam_, Apr 12 2008
%E A139263 a(10)-a(30) from _Petros Hadjicostas_, Jun 02 2020