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A128718
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UU's (doublerises) (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 a path is defined to be the number of its steps.
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2
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1, 1, 2, 1, 5, 4, 1, 9, 18, 8, 1, 14, 50, 56, 16, 1, 20, 110, 220, 160, 32, 1, 27, 210, 645, 840, 432, 64, 1, 35, 364, 1575, 3150, 2912, 1120, 128, 1, 44, 588, 3388, 9534, 13552, 9408, 2816, 256, 1, 54, 900, 6636, 24822, 49644, 53088, 28800, 6912, 512, 1, 65, 1320
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| T(n,0)=1. T(n,1)=(n-1)(n+2)/2=A000096(n-1) T(n,k)=A126182(n,n-k), i.e. triangle is mirror image of A126182. Row sums yield A002212. Sum(k*T(n,k),k=0..n-1)=A128743(n).
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LINKS
| E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
| T(n,k)=binom(n,k)*Sum[binom(k,j)*binomial(n-k+j,j+1), j=0..k]/n (1<=k<=n). T(n,0)=1. G.f.=G-1, where G=G(t,z) satisfies G=1 + tzG^2 + zG - tz.
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EXAMPLE
| T(3,2)=4 because we have UUUDDD, UUUDLD, UUUDDL and UUUDLL.
Triangle starts:
1;
1,2;
1,5,4;
1,9,18,8;
1,14,50,56,16;
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MAPLE
| T:=proc(n, k) if k=0 then 1 else binomial(n, k)*sum(binomial(k, j)*binomial(n-k+j, j+1), j=0..k)/n fi end: for n from 1 to 11 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A002212, A000096, A126182, A128743.
Sequence in context: A141506 A194682 A056242 * A112358 A126351 A157011
Adjacent sequences: A128715 A128716 A128717 * A128719 A128720 A128721
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2007
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