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Search: id:A109196
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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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3
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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; listen)
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OFFSET
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2,2
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COMMENT
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a(n)=sum(k*A109195(n,k),k=0..floor(n/2)). a(n)=(1/2)A109194(n).
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FORMULA
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G.f.=[1-z-sqrt(1-2z-3z^2)]/[2(1-2z-3z^2)].
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EXAMPLE
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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
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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
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Cf. A109195, A109194.
Sequence in context: A025181 A004054 A068995 this_sequence A032637 A034576 A125672
Adjacent sequences: A109193 A109194 A109195 this_sequence A109197 A109198 A109199
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 22 2005
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