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Search: id:A108444
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| A108444 |
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Number of triple descents (i.e. ddd's) 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). |
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+0
1
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| 5, 73, 857, 9505, 103341, 1114969, 11996209, 128989249, 1387480981, 14937170089, 160978217225, 1736820843233, 18760031574077, 202856430706617, 2195832009812065, 23792481053343361, 258038743598973477
(list; graph; listen)
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OFFSET
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2,1
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COMMENT
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a(n)=sum(k*A108443(n,k),k=1..2n-1). Example: a(3)=1*24+2*15+3*3+4*1=73.
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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(2A^2-2zA^2-zA-2)/(1-2zA-3zA^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)=5 because in the ten paths udud, udUdd, uudd, uU(ddd), Uddud, UddUdd, Ududd, UdU(ddd), Uu(ddd) and UU(d[dd)d] (see A027307) we have 5 ddd's (shown between parentheses).
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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*z*A^2-2+2*A^2)/(1-3*z*A^2-2*z*A): Gser:=series(G, z=0, 26): seq(coeff(Gser, z^n), n=2..21);
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CROSSREFS
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Cf. A027307, A108443.
Sequence in context: A070530 A059017 A099667 this_sequence A155662 A159509 A127167
Adjacent sequences: A108441 A108442 A108443 this_sequence A108445 A108446 A108447
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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 10 2005
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