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Search: id:A058987
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| A058987 |
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Catalan(n) - Motzkin(n-1). |
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+0
1
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| 0, 1, 3, 10, 33, 111, 378, 1303, 4539, 15961, 56598, 202214, 727389, 2632605, 9581211, 35047098, 128791323, 475281921, 1760726808, 6545921136, 24415415001, 91340016081, 342658850427, 1288774386909, 4858753673655, 18358309669651
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Number of Dyck paths with a "small Capital N" (a rise then a fall then a rise) - this follows from the exercise on p. 238 of Stanley stating that Motzkin numbers equal to the ballot number without (1,-1,1). Since Ballot numbers are Catalan numbers, the result follows from the well-known bijection with Dyck paths.
a(n+1)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=Catalan(k+1) for k=0,1,...,n. - Michael Somos, Oct 07 2003
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; cf. p. 238.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=1,...,100
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PROGRAM
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(PARI) a(n)=if(n<1, 0, subst(polinterpolate(vector(n, k, binomial(2*k, k)/(k+1))), x, n+1))
(PARI) { allocatemem(932245000); for (n = 1, 100, a=if(n<=1, 0, subst(polinterpolate(vector(n-1, k, binomial(2*k, k)/(k+1))), x, n)); write("b058987.txt", n, " ", a); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 24 2009]
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CROSSREFS
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A000108(n) - A001006(n-1).
Sequence in context: A113299 A126931 A071722 this_sequence A001558 A111639 A149029
Adjacent sequences: A058984 A058985 A058986 this_sequence A058988 A058989 A058990
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KEYWORD
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nonn
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AUTHOR
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You Seng Peng (giawgwan(AT)single.url.com.tw), Jan 17 2001
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