0,2

If A(x)=sum_{k=1..inf} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2) (conjecture).

If A(x)=sum_{k=1..inf} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k)=n^(2k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2). - Emeric Deutsch, Dec 10 2002

A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n -(n^2)*x*A(x)^(n+1) = 1 reduces to A(x)=1+xA(x)^2. - Emeric Deutsch, Dec 10 2002

a(n)=3^(2n)*binomial(4n/3-2/3, n)/(n+1) - EmericDeutsch(AT)msn.com (deutsch(AT)duke.poly.edu), Dec 10 2002

Sequence with offset 1 is expansion of reversion of g.f. x*(1-9*x)^(1/3), which equals x times the g.f. of A004990.

A(x)^3 - 9x*A(x)^4 = 1 since A(x)^3 = 1 +9x +108x^2 +1458x^3 +21060x^4 +... and A(x)^4 = 1 +12x +162x^2 +2340x^3 +... also a(2)=3^3, a(5)=3^10.

Cf. A078531, A078533, A078534, A078535.

Cf. A004990.

Sequence in context: A098401 A011544 A127503 this_sequence A153853 A067000 A157089

Adjacent sequences: A078529 A078530 A078531 this_sequence A078533 A078534 A078535

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 28 2002