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Search: id:A109194
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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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3
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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; listen)
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OFFSET
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2,1
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COMMENT
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a(n)=sum(k*A109193(n,k),k=0..floor(n/2)). a(n)=2*A109196(n).
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FORMULA
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G.f.=[1-z-sqrt(1-2z-3z^2)]/(1-2z-3z^2).
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EXAMPLE
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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
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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
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Cf. A109193, A109196.
Sequence in context: A027561 A126171 A002839 this_sequence A014334 A107239 A106434
Adjacent sequences: A109191 A109192 A109193 this_sequence A109195 A109196 A109197
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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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