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.

1, 1, 3, 15, 99, 773, 6743, 63591, 635307, 6634599, 71759983, 798563065, 9098321475, 105733563393, 1249676348391, 14986826364311, 182027688352427, 2235713532561779, 27732857308708571, 347064951865766607

Offset: 0

Author

Paul D. Hanna, Nov 23 2009

Keywords

• nonn

Comments

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

Examples

			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 +...
		

Links

• Philippe Biane, Patrick Dehornoy, Dual Garside structure of braids and free cumulants of products, arXiv:1407.1604 [math.CO], (7-July-2014)

Crossrefs

Cf. A006664, A001246, A000108.

Cf. A168450 (variant). [From Paul D. Hanna, Nov 29 2009]

Programs

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 *)

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)}

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.

Extensions

Typo in formula corrected by Paul D. Hanna, Nov 24 2009