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 A168344 G.f. A(x) satisfies: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = g.f. of A006664, which is the number of irreducible systems of meanders. 9
 1, 1, 3, 15, 99, 773, 6743, 63591, 635307, 6634599, 71759983, 798563065, 9098321475, 105733563393, 1249676348391, 14986826364311, 182027688352427, 2235713532561779, 27732857308708571, 347064951865766607 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of b^* n-strand braids of length at most 2, see the Biane/Dehornoy reference. - Joerg Arndt, Jul 08 2014 LINKS Philippe Biane, Patrick Dehornoy, Dual Garside structure of braids and free cumulants of products, arXiv:1407.1604 [math.CO], (7-July-2014) FORMULA G.f.: A(x) = F(x/A(x)) where A(x*F(x)) = F(x) = g.f. of A001246, which is the squares of Catalan numbers. G.f.: A(x) = x/Series_Reversion(x*F(x)) where F(x) = g.f. of A001246. G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) = g.f. of A006664. EXAMPLE G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 99*x^4 + 773*x^5 + 6743*x^6 +... A(x) satisfies: A(x*F(x)) = F(x) = g.f. of A001246: F(x) = 1 + x + 4*x^2 + 25*x^3 + 196*x^4 + 1764*x^5 + 17424*x^6 +...+ A000108(n)^2*x^n +... A(x) satisfies: A(x/G(x)) = G(x) = g.f. of A006664: G(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2546*x^6 +... MATHEMATICA F[x_] = (Hypergeometric2F1[-1/2, -1/2, 1, 16x] - 1)/(4x); A[x_] = x/InverseSeries[x F[x] + O[x]^21, x]; CoefficientList[A[x], x] (* Jean-François Alcover, Jul 21 2018, from 2nd formula *) PROG (PARI) {a(n)=local(C_2=vector(n+1, m, (binomial(2*m-2, m-1)/m)^2)); polcoeff(x/serreverse(x*Ser(C_2)), n)} CROSSREFS Cf. A006664, A001246, A000108. Cf. A168450 (variant). [From Paul D. Hanna, Nov 29 2009] Sequence in context: A199416 A046635 A208426 * A091713 A156106 A111546 Adjacent sequences:  A168341 A168342 A168343 * A168345 A168346 A168347 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 23 2009 EXTENSIONS Typo in formula corrected by Paul D. Hanna, Nov 24 2009 STATUS approved

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Last modified August 3 05:29 EDT 2020. Contains 336197 sequences. (Running on oeis4.)