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A102402
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2.
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4
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1, 1, 1, 1, 2, 3, 6, 6, 2, 17, 15, 10, 46, 51, 30, 5, 128, 175, 91, 35, 372, 568, 336, 140, 14, 1109, 1827, 1296, 504, 126, 3349, 5980, 4785, 2010, 630, 42, 10221, 19833, 17215, 8415, 2640, 462, 31527, 66078, 61908, 34210, 11385, 2772, 132, 98178, 220649, 223444, 134706, 50908, 13299, 1716
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| T(n,k) is the number of Lukasiewicz paths of length n having k steps (1,1). A Lukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: T(3,0)=2 because we have HHH and U(2)DD, where H=(1,0), U(2)=(1,2) and D=(1,-1). Row n has 1+floor(n/2) terms. Row sums yield the Catalan numbers (A000108). T(2n,n)=A000108(n). Column 0 is A102403
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FORMULA
| G.f.=G=G(t, z) satisfies z^3*(1-t)G^3+z(1-z+tz)G^2-G+1=0.
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EXAMPLE
| T(4,2)=2 because we have UUDDUUDD and UUDUUDDD, where U=(1,1) and D=(1,-1).
Triangle begins:
1;
1;
1,1;
2,3;
6,6,2;
17,15,10;
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CROSSREFS
| Cf. A000108, A102403.
Sequence in context: A124655 A066838 A084678 * A124498 A197334 A113399
Adjacent sequences: A102399 A102400 A102401 * A102403 A102404 A102405
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 06 2005
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