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%I #31 Jul 17 2020 16:30:41
%S 1,4,20,104,552,2972,16172,88720,489872,2719028,15157188,84799992,
%T 475894200,2677788492,15102309468,85347160608,483183316512,
%U 2739851422820,15558315261812,88462135512712,503569008273992,2869602773253884,16368396446913420,93449566652932784,533954950648248752
%N Number of horizontal segments in all Schroeder paths of length 2n (a horizontal segment is a maximal string of horizontal steps).
%C A Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318).
%H Vincenzo Librandi, <a href="/A104550/b104550.txt">Table of n, a(n) for n = 1..200</a>
%F G.f.: (1-x)*(1-x-sqrt(1-6*x+x^2))/(2*sqrt(1-6*x+x^2)).
%F a(n) = Jacobi_P(n+1,-1,-2,3). [_Paul Barry_, Sep 27 2009]
%F Recurrence: n*a(n) = (7*n-6)*a(n-1) - (7*n-22)*a(n-2) + (n-4)*a(n-3). - _Vaclav Kotesovec_, Oct 17 2012
%F a(n) ~ sqrt(6*sqrt(2)-8)*(3+2*sqrt(2))^n/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 17 2012
%F a(n) = Hyper2F1([-n, n-1], [1], -1). - _Peter Luschny_, Aug 02 2014
%F a(n) = Sum_{k=0..n+1} C(n+1,k)*C(n+k-1,k). - _Vladimir Kruchinin_, Jun 15 2020
%e a(2)=4 because we have (HH),(H)UD,UD(H),U(H)D,UDUD and UUDD; the 4 horizontal segments are shown between parentheses.
%p G:=(1-z)*(1-z-sqrt(1-6*z+z^2))/2/sqrt(1-6*z+z^2): Gser:=series(G,z=0,28): seq(coeff(Gser,z^n),n=1..24);
%p a := n -> hypergeom([-n, n-1], [1], -1);
%p seq(round(evalf(a(n),36)),n=1..23); # _Peter Luschny_, Aug 02 2014
%t Rest[CoefficientList[Series[(1-x)*(1-x-Sqrt[1-6*x+x^2])/ (2*Sqrt[1 -6*x+x^2]), {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%o (PARI) x='x+O('x^66); Vec((1-x)*(1-x-sqrt(1-6*x+x^2))/(2*sqrt(1-6*x+x^2))) \\ _Joerg Arndt_, May 13 2013
%o (Maxima) a(n):=sum(binomial(n+1,k)*binomial(n+k-1,k),k,0,n+1) /* _Vladimir Kruchinin_, Jun 15 2020 */
%Y Cf. A006318, A104549.
%Y Cf. A035028. - _R. J. Mathar_, Aug 28 2008
%K nonn
%O 1,2
%A _Emeric Deutsch_, Mar 14 2005
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