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A109194
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Number of returns to the x-axis (i.e. d or u 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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2, 6, 22, 70, 224, 700, 2174, 6702, 20572, 62920, 191932, 584220, 1775258, 5386846, 16326734, 49435150, 149557436, 452133880, 1366012832, 4124825872, 12449394278, 37558361290, 113266431860, 341467468420, 1029119688014
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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COMMENTS
| a(n)=sum(k*A109193(n,k),k=0..floor(n/2)). a(n)=2*A109196(n).
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FORMULA
| G.f.=[1-z-sqrt(1-2z-3z^2)]/(1-2z-3z^2).
a(n)=2*sum(k=1..n, sum(j=0..n, binomial(j,-n-2*k+2*j)*binomial(n,j))), n>1. [From Vladimir Kruchinin, Oct 12 2011]
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EXAMPLE
| a(3)=6 because we have the following 7 (=A002426(3)) Grand Motzkin paths of length 3: hhh, hu(d), hd(u), u(d)h, d(u)h, uh(d) and dh(u); they have a total of 6 returns to the x-axis (shown between parentheses).
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MAPLE
| g:=(1-z-sqrt(1-2*z-3*z^2))/(1-2*z-3*z^2): gser:=series(g, z=0, 30): seq(coeff(gser, z^n), n=2..28);
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CROSSREFS
| Cf. A109193, A109196.
Sequence in context: A027561 A126171 A002839 * A014334 A107239 A148496
Adjacent sequences: A109191 A109192 A109193 * A109195 A109196 A109197
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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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