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A109196
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Number of returns to the x-axis from above (i.e. d steps hitting the x-axis) in all Grand Motzkin paths of length n. (A Grand Motzkin path of length n is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).).
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2
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1, 3, 11, 35, 112, 350, 1087, 3351, 10286, 31460, 95966, 292110, 887629, 2693423, 8163367, 24717575, 74778718, 226066940, 683006416, 2062412936, 6224697139, 18779180645, 56633215930, 170733734210, 514559844007, 1550364293145
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| a(n)=sum(k*A109195(n,k),k=0..floor(n/2)). a(n)=(1/2)A109194(n).
The substitution x->x/(1+x+x^2), the inverse Motzkin transform, yields a g.f. for the sequence 0,0,2,2,6,4,.., that is 0 followed by 2*A026741(n-1). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 10 2008]
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FORMULA
| G.f.=[1-z-sqrt(1-2z-3z^2)]/[2(1-2z-3z^2)].
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EXAMPLE
| a(3)=3 because we have the following 7 (=A002426(3)) Grand Motzkin paths of length 3: hhh, hu(d), hdu, u(d)h, duh, uh(d) and dhu; they have a total of 3 returns from above to the x-axis (shown between parentheses).
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MAPLE
| g:=(1-z-sqrt(1-2*z-3*z^2))/2/(1-2*z-3*z^2): gser:=series(g, z=0, 32): seq(coeff(gser, z^n), n=2..30);
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CROSSREFS
| Cf. A109195, A109194.
Sequence in context: A025181 A004054 A068995 * A032637 A034576 A125672
Adjacent sequences: A109193 A109194 A109195 * A109197 A109198 A109199
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 22 2005
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