A089387 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.

1, 1, 2, 8, 36, 164, 764, 3652, 17852, 88868, 449004, 2296692, 11870316, 61897140, 325239036, 1720415268, 9154052700, 48961321604, 263092909004, 1419630359572, 7689097400588, 41788586179988, 227819374037340, 1245545102522948

Offset: 0

Author

Emeric Deutsch, Dec 28 2003

Keywords

• nonn

Examples

			Example: a(2)=2 because we have HH and UUDD.
		

Crossrefs

Cf. A006318.

Formula

G.f.: (1-z)(1-z-q)/(z(3-3z-q)), where q = sqrt(1-6z+z^2).

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