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%I #16 Jan 03 2023 03:55:35
%S 1,2,5,16,60,245,1051,4660,21174,98072,461330,2197997,10585173,
%T 51443379,251982793,1242734592,6165798680,30754144182,154123971932,
%U 775669589436,3918703613376,19866054609754,101029857327802,515275408644773
%N Number of Schroeder paths of length 2n having no UHD, UHHD, UHHHD, ..., where U=(1,1), D=(1,-1), H=(2,0).
%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).
%C Equals binomial transform of A119370. - _Paul D. Hanna_, May 17 2006
%H G. C. Greubel, <a href="/A104547/b104547.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A104546(n, 0).
%F G.f.: G = G(z) satisfies G = 1 + z*G + z*G(G - z/(1-z)).
%F G.f.: (1-2*x+2*x^2 - sqrt(1-8*x+16*x^2-12*x^3+4*x^4))/(2*x*(1-x)). - _Paul D. Hanna_, May 17 2006
%F D-finite with recurrence (n+1)*a(n) = 3*(3*n-1)*a(n-1) - 12*(2*n-3)*a(n-2) + 2*(14*n-37)*a(n-3) - 2*(8*n-31)*a(n-4) + 4*(n-5)*a(n-5). - _R. J. Mathar_, Jul 26 2022
%e a(2)=5 because we have HH, HUD, UDH, UDUD and UUDD (UHD does not qualify).
%t CoefficientList[Series[(1-2*x+2*x^2 -Sqrt[1-8*x+16*x^2-12*x^3+4*x^4] )/(2*x*(1-x)), {x,0,40}], x] (* _G. C. Greubel_, Jan 02 2023 *)
%o (PARI) {a(n)=polcoeff(2*(1-x)/(1-2*x+2*x^2 + sqrt(1-8*x+16*x^2-12*x^3+4*x^4+x*O(x^n))),n)} \\ _Paul D. Hanna_, May 17 2006
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-2*x+2*x^2 - Sqrt(1-8*x+16*x^2-12*x^3+4*x^4))/(2*x*(1-x)) )); // _G. C. Greubel_, Jan 02 2023
%o (SageMath)
%o def A104547_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1-2*x+2*x^2 - sqrt(1-8*x+16*x^2-12*x^3+4*x^4))/(2*x*(1-x)) ).list()
%o A104547_list(40) # _G. C. Greubel_, Jan 02 2023
%Y Cf. A006318, A104546, A119370.
%K nonn
%O 0,2
%A _Emeric Deutsch_, Mar 14 2005
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