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Search: id:A114608
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| A114608 |
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Triangle read by rows: T(n,k) is the number of bicolored Dyck paths of semilength n and having k peaks of the form ud (0<=k<=n). A bicolored Dyck path is a Dyck path in which each up-step is of two kinds: u and U. |
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+0
2
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| 1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 45, 96, 66, 16, 1, 197, 501, 450, 170, 25, 1, 903, 2668, 2955, 1520, 365, 36, 1, 4279, 14407, 18963, 12355, 4165, 693, 49, 1, 20793, 78592, 119812, 94528, 41230, 9856, 1204, 64, 1, 103049, 432073, 748548, 693588, 372078, 117054
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Row sums yield A052701. Column 0 yields the little Schroeder numbers (A001003). Sum(k*T(n,k),k=0..n)=A069720(n).
Triangle T(n,k), 0<=k<=n, read by rows; given by [1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)laposte.net), Dec 23 2005
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FORMULA
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T(n, k)=(1/n)binomial(n, k)*sum(2^j*binomial(n, j+1)binomial(n-k, j), j=0..n-k) (k<=n-1); T(n, n)=1. G.f.=G=G(t, z) satisfies G=1+z(G-1+t)G+zG^2.
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EXAMPLE
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T(3,2)=9 because we have (ud)(ud)Ud, (ud)Ud(ud), Ud(ud)(ud), (ud)u(ud)d,
(ud)U(ud)d, u(ud)d(ud), U(ud)d(ud), u(ud)(ud)d, and U(ud)(ud)d (the ud peaks are shown between parentheses).
Triangle starts:
1;
1,1;
3,4,1;
11,19,9,1;
45,96,66,16,1;
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MAPLE
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T:=proc(n, k) if k<=n-1 then (1/n)*binomial(n, k)*sum(2^j*binomial(n, j+1)*binomial(n-k, j), j=0..n-k) elif k=n then 1 else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A052701, A001003, A069720.
Sequence in context: A078068 A054649 A138263 this_sequence A109956 A123319 A076785
Adjacent sequences: A114605 A114606 A114607 this_sequence A114609 A114610 A114611
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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 15 2005
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