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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 Łukasiewicz paths of length n having k level steps at an even level. A Łukasiewicz 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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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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MATHEMATICA
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m = 12; G[_] = 0;
Do[G[z_] = -((G[z]^2 z ((t-1)z - 1)^2 + 1)/((t-1)z^2 + (t-1)z - 1)) + O[z]^m, {m}];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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