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Search: id:A128747
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| A128747 |
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k peaks of height >1 (n>=1; 0<=k<=n-1). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down), and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. |
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| 1, 1, 2, 1, 7, 2, 1, 18, 15, 2, 1, 41, 68, 25, 2, 1, 88, 244, 171, 37, 2, 1, 183, 765, 866, 351, 51, 2, 1, 374, 2199, 3651, 2355, 636, 67, 2, 1, 757, 5954, 13601, 12708, 5421, 1058, 85, 2, 1, 1524, 15438, 46355, 58977, 36198, 11116, 1653, 105, 2, 1, 3059, 38747, 147768
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row sums yield A002212. T(n,0)=1. Sum(k*T(n,k), k=0..n-1)=A128748(n).
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REFERENCES
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E. Deutsch, E. Munarini, and S. Rinaldi, Skew Dyck paths (in preparation).
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FORMULA
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G.f.=G(t,z)=[1-z+zK(t,z)]/[1-zK(t,z)]-1, where K=K(t,z) satisfies zK^2-(1-tz)K+1-z=0 (K is the g.f. for the number of peaks; see A126182).
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EXAMPLE
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T(3,1)=7 because we have UDU(UD)D, UDU(UD)L, U(UD)DUD, UU(UD)DD, UU(UD)LD, UU(UD)DL, and UU(UD)LL (the peaks of height >1 are shown between parentheses).
Triangle starts:
1;
1,2;
1,7,2;
1,18,15,2;
1,41,68,25,2;
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MAPLE
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K:=(1-z*t-sqrt(z^2*t^2-2*z*t+1+4*z^2-4*z))/2/z: G:=z*(2*K-1)/(1-z*K): Gser:=simplify(series(G, z=0, 14)): for n from 1 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A002212, A128748.
Adjacent sequences: A128744 A128745 A128746 this_sequence A128748 A128749 A128750
Sequence in context: A117044 A092666 A019426 this_sequence A124392 A121416 A089329
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KEYWORD
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tabl,nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007
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