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Search: id:A108432
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| A108432 |
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Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1), and have no hills (a hill is either a ud or a Udd starting at the x-axis). |
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+0
4
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| 1, 0, 6, 34, 274, 2266, 19738, 177642, 1640050, 15445690, 147813706, 1433309194, 14052298690, 139063589370, 1387288675002, 13936344557354, 140859338668306, 1431424362057018, 14616361066692778, 149892742974500042
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Column 0 of A108431.
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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. =1/(1+2z-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 (the g.f. of A027307).
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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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g:=1/(1+2*z-z*A-z*A^2): A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3:gser:=series(g, z=0, 27): 1, seq(coeff(gser, z^n), n=1..24);
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CROSSREFS
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Cf. A027307, A108431, A108433.
Sequence in context: A059228 A079568 A063090 this_sequence A125759 A062819 A092336
Adjacent sequences: A108429 A108430 A108431 this_sequence A108433 A108434 A108435
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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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