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Search: id:A114709
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| A114709 |
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Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k horizontal steps on the x-axis (0<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1. |
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+0
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| 1, 0, 1, 2, 0, 1, 6, 4, 0, 1, 26, 12, 6, 0, 1, 114, 56, 18, 8, 0, 1, 526, 252, 90, 24, 10, 0, 1, 2502, 1192, 414, 128, 30, 12, 0, 1, 12194, 5772, 2006, 600, 170, 36, 14, 0, 1, 60570, 28536, 9882, 2976, 810, 216, 42, 16, 0, 1, 305526, 143388, 49554, 14904, 4110, 1044
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Row sums are the little Schroeder numbers (A001003). Column 0 is A114710. Sum(k*T(n,k),k=0..n)=A010683(n-1).
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FORMULA
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G.f.=1/(1+z-tz-zR), where R=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).
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EXAMPLE
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T(4,2)=6 because we have (HH)UHD,(HH)UUDD,(H)UHD(H),(H)UUDD(H),UHD(HH) and
UUDD(HH), where U=(1,1), D=(1,-1) and H=(2,0) (the H's on the x-axis are shown between parentheses).
Triangle starts:
1;
0,1;
2,0,1;
6,4,0,1;
26,12,6,0,1;
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MAPLE
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R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/(1+z-t*z-z*R): Gser:=simplify(series(G, z=0, 14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 10 do seq(coeff(t*P[n], t^j), j=1..n+1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001003, A114710, A010683.
Sequence in context: A021500 A059299 A128722 this_sequence A089949 A085845 A138106
Adjacent sequences: A114706 A114707 A114708 this_sequence A114710 A114711 A114712
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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 26 2005
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