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A108440
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Triangle read by rows: T(n,k) is 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 k u=(2,1) steps among the steps leading to the first d step.
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0
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1, 1, 1, 5, 4, 1, 33, 25, 7, 1, 249, 184, 54, 10, 1, 2033, 1481, 446, 92, 13, 1, 17485, 12620, 3863, 846, 139, 16, 1, 156033, 111889, 34637, 7881, 1411, 195, 19, 1, 1431281, 1021424, 318812, 74492, 14102, 2168, 260, 22, 1, 13412193, 9536113, 2995228
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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REFERENCES
| Problem 10658, American Math. Monthly, 107, 2000, 368-370.
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FORMULA
| G.f.=G=G(t, z)=1/(1-tzA-zA^2)-1, 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
| T(2,1)=4 because we have udud, udUdd, uUddd and Uuddd.
Triangle begins:
.1;
.1,1;
.5,4,1;
.33,25,7,1;
.249,184,54,10,1;
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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/(1-t*z*A-z*A^2): Gser:=simplify(series(G, z=0, 12)): P[0]:=1: for n from 1 to 9 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 9 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form
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CROSSREFS
| Row sums yield A027307. Column 0 yields A034015.
Cf. A027307, A034015, A108441.
Sequence in context: A008955 A182824 A152862 * A102220 A201680 A109430
Adjacent sequences: A108437 A108438 A108439 * A108441 A108442 A108443
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 08 2005
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