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Search: id:A101920
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| A101920 |
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Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at an odd height. |
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+0
2
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| 2, 5, 1, 13, 8, 1, 34, 42, 13, 1, 89, 183, 102, 19, 1, 233, 717, 624, 205, 26, 1, 610, 2622, 3275, 1650, 366, 34, 1, 1597, 9134, 15473, 11020, 3716, 602, 43, 1, 4181, 30691, 67684, 64553, 30520, 7483, 932, 53, 1, 10946, 100284, 279106, 342867, 215481
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis (Schroeder paths are counted by the large Schroeder numbers, A006318). Row sums are the large Schroeder numbers (A006318). Column 0 yields the odd-indexed Fibonacci numbers (A001519).
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FORMULA
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G.f.=G=G(t, z) satisfies tz(1-z)G^2-(1-3z+tz+z^2)G+1-z=0.
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EXAMPLE
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T(3,1)=8 because we have HUU'DD, UDUU'DD, UU'DDH, UU'DDUD, UHU'DD, UU'DHD, UU'HDD, and UU'UDDD, the up steps starting at odd heights being shown with the prime sign.
Triangle begins:
2;
5,1;
13,8,1;
34,42,13,1;
89,183,102,19,1;
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MAPLE
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G := 1/2/(-t*z+t*z^2)*(-1+3*z-t*z-z^2+sqrt(1-6*z-2*t*z+11*z^2+2*t*z^2-6*z^3+t^2*z^2-2*t*z^3+z^4)): Gser:=simplify(series(G, z=0, 13)):for n from 1 to 11 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 11 do seq(coeff(t*P[n], t^k), k=1..n) od; # yields the sequence in triangular form
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CROSSREFS
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Cf. A006318, A001519, A101919.
Sequence in context: A085358 A120235 A089618 this_sequence A114494 A118964 A073187
Adjacent sequences: A101917 A101918 A101919 this_sequence A101921 A101922 A101923
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 20 2004
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