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Search: id:A128095
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| A128095 |
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Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having k steps that touch the x-axis (1<=k<=n). |
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2
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| 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 2, 2, 3, 1, 0, 4, 4, 4, 4, 1, 0, 8, 8, 8, 7, 5, 1, 0, 17, 16, 17, 14, 11, 6, 1, 0, 37, 34, 36, 31, 23, 16, 7, 1, 0, 82, 74, 79, 68, 53, 36, 22, 8, 1, 0, 185, 164, 177, 152, 121, 86, 54, 29, 9, 1, 0, 423, 370, 402, 346, 278, 204, 134, 78, 37, 10, 1, 0
(list; table; graph; listen)
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OFFSET
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1,9
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COMMENT
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T(n,k)=number of secondary structures of size n in which the shortest path from one end to the other one has length k-1. Row sums yield A004148. T(n,2)=A004148(n-2). T(n,3)=2*A004148(n-3) for n>=4. Sum(k*T(n,k),k=1..n)=A128096(n).
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FORMULA
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G.f.=2/[2-2tz-t^2+t^2*z+t^2*z^2+t^2*sqrt((1+z+z^2)(1-3z+z^2))]-1.
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EXAMPLE
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T(5,4)=3 because we have HU(H)DH, HHU(H)D, and U(H)DHH, where U=(1,1), H=(1,0), and D=(1,-1) and the steps that do not touch the x-axis are shown between parentheses.
Triangle starts:
1;
0,1;
0,1,1;
0,1,2,1;
0,2,2,3,1;
0,4,4,4,4,1;
0,8,8,8,7,5,1;
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MAPLE
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G:=2/(2-2*t*z-t^2+t^2*z+t^2*z^2+t^2*sqrt((1+z+z^2)*(1-3*z+z^2)))-1: Gser:=simplify(series(G, z=0, 15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 13 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A004148, A128096.
Adjacent sequences: A128092 A128093 A128094 this_sequence A128096 A128097 A128098
Sequence in context: A047654 A058487 A062243 this_sequence A097854 A019591 A091967
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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), Feb 14 2007
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