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A092107
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UUU's (triple rises) where U=(1,1). Rows have 1,1,1,2,3,4,5,... entries, respectively.
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2
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1, 1, 2, 4, 1, 9, 4, 1, 21, 15, 5, 1, 51, 50, 24, 6, 1, 127, 161, 98, 35, 7, 1, 323, 504, 378, 168, 48, 8, 1, 835, 1554, 1386, 750, 264, 63, 9, 1, 2188, 4740, 4920, 3132, 1335, 390, 80, 10, 1, 5798, 14355, 17028, 12507, 6237, 2200, 550, 99, 11, 1, 15511, 43252, 57816
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Column 0 gives the Motzkin numbers (A001006), column 1 gives A014532. Row sums are the Catalan numbers (A000108).
Equal to A171380*B (without the zeros), B = A007318 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2009]
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LINKS
| A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
Yidong Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 287 (2004), 177-186.
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FORMULA
| G.f. G=G(t, z) satisfies z(t+z-tz)G^2-(1-z+tz)G+1=0.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A001405(n), A001006(n), A000108(n), A033321(n) for x = -1, 0, 1, 2 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2009]
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EXAMPLE
| T(5,2)=5 because we have (U[UU)U]DUDDDD, (U[UU)U]DDUDDD, (U[UU)U]DDDUDD,
(U[UU)U]DDDDUD and UD(U[UU)U]DDDD, where U=(1,1), D=(1,-1); the triple rises are shown between parentheses.
[1],[1],[2],[4, 1],[9, 4, 1],[21, 15, 5, 1],[51, 50, 24, 6, 1],[127, 161, 98, 35, 7, 1]
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CROSSREFS
| Cf. A001006, A014532.
Cf. A000108, A001006, A001405, A007318, A033321, A171380 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2009]
Sequence in context: A174135 A182903 A169840 * A114489 A101974 A097607
Adjacent sequences: A092104 A092105 A092106 * A092108 A092109 A092110
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2004
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