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Search: id:A114276
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| A114276 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having length of second ascent equal to k (0<=k<=n-1). |
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+0
2
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| 1, 1, 1, 1, 3, 1, 1, 8, 4, 1, 1, 22, 13, 5, 1, 1, 64, 41, 19, 6, 1, 1, 196, 131, 67, 26, 7, 1, 1, 625, 428, 232, 101, 34, 8, 1, 1, 2055, 1429, 804, 376, 144, 43, 9, 1, 1, 6917, 4861, 2806, 1377, 573, 197, 53, 10, 1, 1, 23713, 16795, 9878, 5017, 2211, 834, 261, 64, 11, 1, 1
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Column 1 yields A014138, column 2 yields A001453, column 3 yields A114277. Row sums are the Catalan numbers (A000108).
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FORMULA
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T(n, k)=(k+1)*sum(binomial(2*n-k+1-2*j, n-j+1)/(2*n-k-2*j+1), j=1..n-k) if 1<=k<=n-1; T(n, 0)=1. G.f. = (1-tz)/[(1-z)(1-tzC)]-1 where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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T(4,2)=4 because we have UD(UU)DDUD, UD(UU)DUDD, UUD(UU)DDD, and UUDD(UU)DD (second ascent shown between parentheses).
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MAPLE
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T:=proc(n, k) if k=0 then 1 elif k<=n-1 then (k+1)*sum(binomial(2*n-k+1-2*j, n-j+1)/(2*n-k-2*j+1), j=1..n-k) else 0 fi end: for n from 1 to 12 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000108, A014138, A001453, A114277.
Sequence in context: A131932 A016462 A121461 this_sequence A098747 A122897 A117425
Adjacent sequences: A114273 A114274 A114275 this_sequence A114277 A114278 A114279
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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), Nov 20 2005
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