_Paul D. Hanna (pauldhanna(AT)juno.com), _, Nov 23 2009

Self-convolution of A006664, which is the number of irreducible systems of meanders.

1, 2, 5, 20, 112, 768, 5984, 50856, 460180, 4366076, 42988488, 436066232, 4532973676, 48095557700, 519247705968, 5690272928520, 63172884082028, 709373555125356, 8046263496489260, 92089662771965492, 1062482514810065752

0,2

G.f.: A(x) = x/Series_Reversion(x*F(x)^2) where F(x) = g.f. of A001246, which is the squares of Catalan numbers.

G.f.: A(x) = F(x/A(x))^2 where A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A001246.

G.f.: A(x) = 1 + 2*x + 5*x^2 + 20*x^3 + 112*x^4 + 768*x^5 +...

A(x)^(1/2) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2546*x^6 +...+ A006664(n)*x^n +...

G.f. satisfies: A(x*F(x)^2) = F(x)^2 where 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 +...

F(x)^2 = 1 + 2*x + 9*x^2 + 58*x^3 + 458*x^4 + 4120*x^5 + 40569*x^6 +...+ A168358(n)*x^n +...

(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)^2), n)}

Cf. A168358, A168344, A001246, A006664, A000108.

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 23 2009

approved