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Search: id:A108450
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| A108450 |
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Number of pyramids in all 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) (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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+0
3
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| 2, 10, 58, 402, 3122, 26010, 227050, 2049186, 18964194, 178976426, 1715905050, 16665027378, 163611970066, 1621103006010, 16189480081354, 162791835045698, 1646810150270914, 16748008972020554, 171135004105459194
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OFFSET
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1,1
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COMMENT
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A108450(n)=sum(k*A108445(k),k=1..n) (for example, A108450(3)=1*18+2*8+3*8=58). A108450(n)=2*A108453(n). A108450 =2*partial sums of A032349.
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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.=2zA^2/(1-z), 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
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a(2)=10 because in the A027307(2)=10 paths we have alltogether 10 pyramids (shown between parentheses): (ud)(ud), (ud)(Udd), (uudd), uUddd, (Udd)(ud), (Udd)(Udd), Ududd, UdUddd, Uuddd, (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:=2*z*A^2/(1-z): gser:=series(g, z=0, 25): seq(coeff(gser, z^n), n=1..22);
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CROSSREFS
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Cf. A027307, A108445, A108453, A032349.
Sequence in context: A075870 A074608 A086871 this_sequence A112369 A124964 A026132
Adjacent sequences: A108447 A108448 A108449 this_sequence A108451 A108452 A108453
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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 11 2005
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