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Search: id:A128098
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| A128098 |
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Number of steps that touch the x-axis in all Motzkin paths of length n. |
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+0
2
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| 1, 4, 11, 30, 80, 214, 574, 1548, 4197, 11440, 31339, 86252, 238407, 661584, 1842585, 5148960, 14432643, 40569804, 114339777, 323031750, 914683602, 2595411126, 7378861196, 21016701652, 59962687675, 171353419536, 490407962229
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n)=Sum(k*A128097(n,k), k=1..n).
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FORMULA
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G.f.=4[1-sqrt(1-2z-3z^2)]/[1-z+sqrt(1-2z-3z^2)]^2.
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EXAMPLE
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a(3)=11 because in the Motzkin paths of length 3 (namely HHH, HUD, UDH, and UHD, where H=(1,0), U=(1,1), and D=(1,-1)) all the steps, with the exception of H in UHD, touch the x-axis.
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MAPLE
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g:=4*(1-sqrt(1-2*z-3*z^2))/(1-z+sqrt(1-2*z-3*z^2))^2: gser:=series(g, z=0, 35): seq(coeff(gser, z, n), n=1..32);
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CROSSREFS
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Cf. A128097.
Sequence in context: A026583 A110034 A114726 this_sequence A019495 A019496 A021006
Adjacent sequences: A128095 A128096 A128097 this_sequence A128099 A128100 A128101
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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), Feb 16 2007
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