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A108441
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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=(1,2) steps among the steps leading to the first d step.
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2
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1, 1, 1, 3, 6, 1, 15, 39, 11, 1, 97, 284, 100, 16, 1, 721, 2249, 888, 186, 21, 1, 5827, 18890, 7977, 1952, 297, 26, 1, 49759, 165519, 72991, 19731, 3601, 433, 31, 1, 441729, 1496696, 680096, 196864, 40586, 5960, 594, 36, 1, 4035937, 13865297, 6439656
(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-zA-tzA^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)=6 because we have uUddd, Uddud, UddUdd, Ududd, UdUddd and Uuddd.
Triangle begins:
1;
1,1;
3,6,1;
15,39,11,1;
97,284,100,16,1;
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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-z*A-t*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 A108442.
Sequence in context: A100960 A130852 A138799 * A176231 A176230 A094445
Adjacent sequences: A108438 A108439 A108440 * A108442 A108443 A108444
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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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