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A089387
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Number of Schroeder paths of semilength n (i.e., lattice paths from (0,0) to (2n,0), with steps H=(2,0), U=(1,1) and D(1,-1) and not going below the x-axis) with no UD, UHD, UHHD, UHHHD, ... starting at level zero.
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%I #7 Jul 24 2022 12:05:57
%S 1,1,2,8,36,164,764,3652,17852,88868,449004,2296692,11870316,61897140,
%T 325239036,1720415268,9154052700,48961321604,263092909004,
%U 1419630359572,7689097400588,41788586179988,227819374037340,1245545102522948
%N Number of Schroeder paths of semilength n (i.e., lattice paths from (0,0) to (2n,0), with steps H=(2,0), U=(1,1) and D(1,-1) and not going below the x-axis) with no UD, UHD, UHHD, UHHHD, ... starting at level zero.
%F G.f.: (1-z)(1-z-q)/(z(3-3z-q)), where q = sqrt(1-6z+z^2).
%F D-finite with recurrence 2*(n+1)*a(n) +(-17*n+7)*a(n-1) +(37*n-59)*a(n-2) +(-37*n+89)*a(n-3) +(17*n-61)*a(n-4) +2*(-n+5)*a(n-5)=0. - _R. J. Mathar_, Jul 24 2022
%e Example: a(2)=2 because we have HH and UUDD.
%Y Cf. A006318.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Dec 28 2003
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