%I #11 Jul 21 2018 04:22:39

%S 1,1,3,15,99,773,6743,63591,635307,6634599,71759983,798563065,

%T 9098321475,105733563393,1249676348391,14986826364311,182027688352427,

%U 2235713532561779,27732857308708571,347064951865766607

%N 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.

%C Number of b^* n-strand braids of length at most 2, see the Biane/Dehornoy reference. - _Joerg Arndt_, Jul 08 2014

%H Philippe Biane, Patrick Dehornoy, <a href="http://arxiv.org/abs/1407.1604">Dual Garside structure of braids and free cumulants of products</a>, arXiv:1407.1604 [math.CO], (7-July-2014)

%F 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.

%F G.f.: A(x) = x/Series_Reversion(x*F(x)) where F(x) = g.f. of A001246.

%F G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) = g.f. of A006664.

%e G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 99*x^4 + 773*x^5 + 6743*x^6 +...

%e A(x) satisfies: A(x*F(x)) = F(x) = g.f. of A001246:

%e F(x) = 1 + x + 4*x^2 + 25*x^3 + 196*x^4 + 1764*x^5 + 17424*x^6 +...+ A000108(n)^2*x^n +...

%e A(x) satisfies: A(x/G(x)) = G(x) = g.f. of A006664:

%e G(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2546*x^6 +...

%t F[x_] = (Hypergeometric2F1[-1/2, -1/2, 1, 16x] - 1)/(4x);

%t A[x_] = x/InverseSeries[x F[x] + O[x]^21, x];

%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Jul 21 2018, from 2nd formula *)

%o (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)}

%Y Cf. A006664, A001246, A000108.

%Y Cf. A168450 (variant). [From _Paul D. Hanna_, Nov 29 2009]

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 23 2009

%E Typo in formula corrected by _Paul D. Hanna_, Nov 24 2009