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A065601
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Number of Dyck paths of length 2n with exactly 1 hill.
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4
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0, 1, 0, 2, 4, 13, 40, 130, 432, 1466, 5056, 17672, 62460, 222853, 801592, 2903626, 10582816, 38781310, 142805056, 528134764, 1960825672, 7305767602, 27307800400, 102371942932, 384806950624, 1450038737668, 5476570993440, 20727983587220, 78606637060012
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OFFSET
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0,4
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COMMENTS
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Convolution of A000957(n) with itself gives a(n-1).
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LINKS
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FORMULA
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Reference gives g.f.'s.
Conjecture: 2*(n+1)*a(n) +(-3*n+2)*a(n-1) +2*(-9*n+19)*a(n-2) +4*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Dec 10 2013
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MAPLE
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b:= proc(x, y, h, z) option remember;
`if`(x<0 or y<x, 0, `if`(x=0 and y=0, `if`(h, 0, 1),
b(x-1, y, h, is(x=y))+ `if`(h and z, b(x, y-1, false$2),
`if`(z, 0, b(x, y-1, h, false)))))
end:
a:= n-> b(n$2, true$2):
# second Maple program:
series(((1-sqrt(1-4*x))/(3-sqrt(1-4*x)))^2/x, x=0, 30); # Mark van Hoeij, Apr 18 2013
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MATHEMATICA
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CoefficientList[Series[((1-Sqrt[1-4*x])/(3-Sqrt[1-4*x]))^2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
Table[Sum[(-1)^j*(j+1)*(j+2)*Binomial[2*n-1-j, n], {j, 0, n-1}]/(n+1), {n, 0, 30}] (* Vaclav Kotesovec, May 18 2015 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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