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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 23 2009
EXTENSIONS
Typo in formula corrected by Paul D. Hanna, Nov 24 2009
STATUS
approved