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Search: id:A105633
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| 1, 2, 4, 9, 22, 57, 154, 429, 1223, 3550, 10455, 31160, 93802, 284789, 871008, 2681019, 8298933, 25817396, 80674902, 253106837, 796968056, 2517706037, 7977573203, 25347126630, 80738862085, 257778971504, 824798533933
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OFFSET
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0,2
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COMMENT
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Binomial transform of A007477. INVERT transform of A082582. First differences give A086581 and A025242 (offset 1). Is this sequence equal to A057580?
a(n) = the number of Dyck paths of semilength n+1 avoiding UUDU. a(n) = the number of Dyck paths of semilength n+1 avoiding UDUU. Sequence A105633 is the first column of the triangle A116424. E.g. a(2) = 4 because there exist four Dyck paths of semilength 3 that avoid UUDU : UDUDUD, UDUUDD, UUDDUD, UUUDDD, as well as four Dyck paths of semilength 3 that avoid UDUU : UDUDUD, UUDUDD, UUDDUD, UUUDDD. - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
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REFERENCES
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N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.
A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.
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FORMULA
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G.f.: A(x) = (1-x - sqrt((1-x)^2 - 4*x^2/(1-x)))/(2*x^2).
a(n) = 2*a(n-1) + Sum(a(i)*(a(n-1-i)-a(n-2-i)),i=1..n-2). a(n) = Sum((-1)^i * binomial(n+1-i,i) * binomial(2*(n+1)-3*i,n-2*i) /(n+1-i), i=0..[n/2]). - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
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PROGRAM
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(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(2/(1-X)/(1-X+sqrt((1-X)^2-4*X^2/(1-X))), n, x)}
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CROSSREFS
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Cf. A105632, A057580.
Sequence in context: A037245 A130018 A099754 this_sequence A099241 A124380 A059019
Adjacent sequences: A105630 A105631 A105632 this_sequence A105634 A105635 A105636
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Apr 17 2005
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EXTENSIONS
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More terms from I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
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