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Search: id:A114687
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| A114687 |
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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 double rises (0<=k<=n-1). |
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+0
1
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| 1, 1, 2, 1, 6, 4, 1, 12, 24, 8, 1, 20, 80, 80, 16, 1, 30, 200, 400, 240, 32, 1, 42, 420, 1400, 1680, 672, 64, 1, 56, 784, 3920, 7840, 6272, 1792, 128, 1, 72, 1344, 9408, 28224, 37632, 21504, 4608, 256, 1, 90, 2160, 20160, 84672, 169344, 161280, 69120, 11520
(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 are the little Schroeder numbers (A001003). Sum(k*T(n,k),k=0..n-1)=2*A050152(n-1). Mirror image of A114656.
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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^k*binomial(n, k)*binomial(n, k+1)/n. G.f.=G=G(t, z) satisfies G=z(1+G)(1+2tG).
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EXAMPLE
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T(3,2)=4 because we have UbUbUDDD, UbUrUDDD, UrUbUDDD, and UrUrUDDD, where U=(1,1), D=(1,-1) and
b (r) indicates a blue (red) double rise.
Triangle begins:
1;
1,2;
1,6,4;
1,12,24,8;
1,20,80,80,16.
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MAPLE
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T:=(n, k)->2^k*binomial(n, k)*binomial(n, k+1)/n: for n from 1 to 11 do seq(T(n, k), k=0..n-1) od;
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CROSSREFS
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Cf. A001003, A050152, A114656.
Sequence in context: A133166 A051482 A123519 this_sequence A137594 A112360 A118040
Adjacent sequences: A114684 A114685 A114686 this_sequence A114688 A114689 A114690
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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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