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Search: id:A114656
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| A114656 |
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Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; called also marked Dyck paths) of semilength n and having k peaks (1<=k<=n). |
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+0
4
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| 1, 2, 1, 4, 6, 1, 8, 24, 12, 1, 16, 80, 80, 20, 1, 32, 240, 400, 200, 30, 1, 64, 672, 1680, 1400, 420, 42, 1, 128, 1792, 6272, 7840, 3920, 784, 56, 1, 256, 4608, 21504, 37632, 28224, 9408, 1344, 72, 1, 512, 11520, 69120, 161280, 169344, 84672, 20160, 2160, 90
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row sums are the little Schroeder numbers (A001003). Sum(k*T(n,k),k=1..n)=A047781(n). T(n,k)=(1/2)A114655(n,k).
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LINKS
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D. Callan, Polygon Dissections and Marked Dyck Paths
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FORMULA
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T(n, k)=2^(n-k)*binomial(n, k)*binomial(n, k-1)/n. G.f.=G=G(t, z) satisfies G=z(2G+t)(G+1).
T(n,k)=A001263(n,k)*2^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 11 2007
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EXAMPLE
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T(3,2)=6 because we have (UD)Ub(UD)D, (UD)Ur(UD)D, Ub(UD)D(UD), Ur(UD)D(UD), Ub(UD)(UD)D, and Ur(UD)(UD)D, where U=(1,1), D=(1,-1) and
b (r) indicates a blue (red) double rise (the peaks are shown between parentheses).
Triangle begins:
1;
2,1;
4,6,1;
8,24,12,1;
16,80,80,20,1;
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MAPLE
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T:=(n, k)->2^(n-k)*binomial(n, k)*binomial(n, k-1)/n: for n from 1 to 11 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001003, A047781, A114655.
Sequence in context: A121757 A109822 A114192 this_sequence A075497 A079474 A091543
Adjacent sequences: A114653 A114654 A114655 this_sequence A114657 A114658 A114659
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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 23 2005
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