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A108449
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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 having no pyramids (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis).
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3
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1, 0, 4, 32, 252, 2112, 18484, 166976, 1545548, 14583808, 139774180, 1356966240, 13316740764, 131890671680, 1316627340564, 13234192747648, 133829733962732, 1360586260341248, 13898403178004420, 142578916276009632
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Column 0 of A108445.
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REFERENCES
| Problem 10658, American Math. Monthly, 107, 2000, 368-370.
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FORMULA
| G.f.=(1-z)/[1+z-z(1-z)A(1+A)], where A=1+zA^2+zA^3=(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
| a(2)=4 because the paths uUddd, Ududd, UdUddd and Uuddd have no pyramids.
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MAPLE
| A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=(1-z)/(1+z-z*(1-z)*A*(1+A)): gser:=series(g, z=0, 24): 1, seq(coeff(gser, z^n), n=1..21);
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CROSSREFS
| Cf. A027307, A108445.
Sequence in context: A033515 A147551 A007278 * A092811 A013731 A009509
Adjacent sequences: A108446 A108447 A108448 * A108450 A108451 A108452
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 11 2005
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