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Search: id:A108424
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| A108424 |
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Number of paths from (0,0) to (3n,0) that stay in the first quadrant, consist of steps u=(2,1),U=(1,2), or d=(1,-1) and do not touch the x-axis, except at the endpoints. |
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+0
3
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| 2, 6, 34, 238, 1858, 15510, 135490, 1223134, 11320066, 106830502, 1024144482, 9945711566, 97634828354, 967298498358, 9659274283650, 97119829841854, 982391779220482, 9990160542904134, 102074758837531810
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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a(n)=A027307(n-1)+A032349(n)
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REFERENCES
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Problem 10658, American Math. Monthly, 107, 2000, 368-370.
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FORMULA
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G.f.=zA+zA^2, where A=1+zA^2+zA^3 or, equivalently, A=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3.
a(n)=[n*2^n*binomial(2*n, n)/((2n-1)(n+1))]sum([binomial(n-1, j)]^2/[2^j*binomial(n+j+1, j)], j=0..n-1).
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EXAMPLE
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a(2)=6 because we have uudd, uUddd, Ududd, UdUddd, Uuddd, and UUdddd.
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MAPLE
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A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=z*A+z*A^2: Gser:=series(G, z=0, 28): seq(coeff(Gser, z^n), n=1..25);
a:=proc(n) if n=1 then 2 else (n*2^n*binomial(2*n, n)/((2*n-1)*(n+1)))*sum(binomial(n-1, j)^2/2^j/binomial(n+j+1, j), j=0..n-1) fi end: seq(a(n), n=1..19);
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CROSSREFS
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Cf. A027307, A032349.
Sequence in context: A052824 A019029 A019032 this_sequence A002685 A052878 A076863
Adjacent sequences: A108421 A108422 A108423 this_sequence A108425 A108426 A108427
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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 03 2005
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