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A191307
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Sum of the heights of the first peaks in all dispersed Dyck paths of length n (i.e. in Motzkin paths of length n with no (1,0)-steps at positive heights).
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1
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0, 0, 1, 2, 6, 11, 26, 47, 103, 187, 397, 727, 1519, 2806, 5809, 10814, 22254, 41702, 85460, 161042, 329002, 622932, 1269578, 2413644, 4909788, 9367188, 19024888, 36408748, 73850908, 141714823, 287137498, 552320023, 1118042743, 2155201063, 4359162493, 8419091443
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OFFSET
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0,4
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COMMENTS
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a(n)=Sum(k*A191306(n,k), k>=0).
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LINKS
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Table of n, a(n) for n=0..35.
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FORMULA
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G.f.: ((1-z-z^2)*sqrt(1-4*z^2) - (1-2*z)*(1+z-z^2))/(2*z^3*(1-z)*(1-2*z)).
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EXAMPLE
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a(4)=6 because, denoting U=(1,1), D=(1,-1), H=(1,0), in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD the sum of the heights of the first peaks is 0+1+1+1+1+2=6.
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MAPLE
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g:=(((1-z-z^2)*sqrt(1-4*z^2)-(1-2*z)*(1+z-z^2))*1/2)/(z^3*(1-z)*(1-2*z)): gser:=series(g, z=0, 40): seq(coeff(gser, z, n), n=0..35);
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CROSSREFS
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Cf. A191306
Sequence in context: A079118 A211054 A034466 * A007186 A033304 A091622
Adjacent sequences: A191304 A191305 A191306 * A191308 A191309 A191310
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, May 30 2011
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STATUS
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approved
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