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A102404
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Triangle read by rows: T(n,k) is number of Dyck paths of semilength n having k ascents of length 1 starting at an even level.
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2
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1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 5, 5, 3, 0, 1, 14, 14, 9, 4, 0, 1, 39, 46, 27, 14, 5, 0, 1, 114, 143, 101, 44, 20, 6, 0, 1, 339, 466, 341, 184, 65, 27, 7, 0, 1, 1028, 1524, 1212, 664, 300, 90, 35, 8, 0, 1, 3163, 5043, 4279, 2539, 1145, 454, 119, 44, 9, 0, 1, 9852, 16812, 15206, 9564, 4665, 1819, 651, 152, 54, 10, 0, 1
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OFFSET
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0,7
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COMMENTS
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T(n,k) is number of Lukasiewicz paths of length n having k level steps at an even level. 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(4,1)=5 because we have (H)UHD, (H)U(2)DD, UHD(H), U(2)DD(H) and U(2)(H)DD, where H=(1,0), U(2)=(1,2) and D=(1,-1) and the level steps at even level are shown between parentheses. Row n contains n+1 terms. Row sums yield the Catalan numbers (A000108). Column 0 is A102406.
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LINKS
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Table of n, a(n) for n=0..77.
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FORMULA
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G.f.=G=G(t, z) satisfies z(1+z-tz)^2*G^2-(1+z+z^2-tz-tz^2)G+1=0.
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EXAMPLE
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T(3,1)=2 because we have (U)DUUDD and UUDD(U)D, where U=(1,1), D=(1,-1) and the ascents of length 1 starting at an even level are shown between parentheses.
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CROSSREFS
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Cf. A000108, A102405, A102406.
Sequence in context: A146326 A158852 A188285 * A089246 A105929 A065600
Adjacent sequences: A102401 A102402 A102403 * A102405 A102406 A102407
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch, Jan 06 2005
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STATUS
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approved
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