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A114693
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Number of returns to the x-axis in all hill-free Schroeder paths of length 2n+4. A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1.
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1
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2, 10, 48, 232, 1138, 5666, 28592, 145984, 752978, 3918282, 20547456, 108482952, 576187554, 3076640898, 16506527392, 88938911296, 481067145826, 2611212958154, 14218923060752, 77653486423528, 425227486222482
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| a(n)=sum(k*A114692(n+2,k),k=0..1+floor(n/2)).
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FORMULA
| G.f.=2[1-3z-sqrt(1-6z+z^2)]/[z(1+z+sqrt(1-6z+z^2)]^2.
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EXAMPLE
| a(0)=2 because in the three hill-free Schroeder paths of length 4, namely HH, UH(D) and UUD(D), we have altogether 2 returns to the x-axis (shown between parentheses).
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MAPLE
| G:=2*(1-3*z-sqrt(1-6*z+z^2))/z^2/(1+z+sqrt(1-6*z+z^2))^2:Gser:=series(G, z=0, 30): 2, seq(coeff(Gser, z^n), n=1..23);
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CROSSREFS
| Cf. A114692.
Sequence in context: A192696 A181296 A065982 * A121950 A086853 A036918
Adjacent sequences: A114690 A114691 A114692 * A114694 A114695 A114696
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2005
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