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

0,3

Comments

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

Links

Table of n, a(n) for n=0..19.

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