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Search: id:A108437
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| A108437 |
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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 height of the first peak equal to k. |
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+0
1
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| 1, 1, 2, 5, 2, 1, 10, 28, 13, 11, 3, 1, 66, 196, 90, 89, 34, 18, 4, 1, 498, 1532, 694, 736, 311, 197, 66, 26, 5, 1, 4066, 12804, 5738, 6344, 2800, 1937, 762, 367, 110, 35, 6, 1, 34970, 111964, 49758, 56576, 25560, 18636, 7953, 4263, 1551, 615, 167, 45, 7, 1
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row n contains 2n terms. Row sums yield A027307. T(n,1)=A027307(n-1).
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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.=G=G(t, z)=1/(1-tzA-t^2*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
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T(2,3)=2 because we have uUddd and Uuddd.
Triangle begins:
1,1;
2,5,2,1;
10,28,13,11,3,1;
66,196,90,89,34,18,4,1;
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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:=1/(1-t*z*A-t^2*z*A^2)-1: Gserz:=simplify(series(G, z=0, 10)): for n from 1 to 9 do P[n]:=sort(coeff(Gserz, z^n)) od: for n from 1 to 9 do seq(coeff(P[n], t^k), k=1..2*n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A027307.
Sequence in context: A064334 A061176 A124780 this_sequence A065291 A065267 A100955
Adjacent sequences: A108434 A108435 A108436 this_sequence A108438 A108439 A108440
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 04 2005
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