0,3

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 2008: (Start)

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then

A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);

thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1. (End)

G.f. satisfies: A(x) = 1 + x*B(x)^2 = 1 + (1 + x*C(x)^2 )^2 where B(x) and C(x) satisfy: C(x) = B(x)*B(A(x)-1), B(x) = A(A(x)-1), B(A(x)-1) = A(B(x)-1), B(x/A(x)^2) = A(x), B(x) = A(x*B(x)^2) and B(x) is g.f. of A120971.

a(n) = [x^(n-1)] (1/n)*A(x)^(2n) for n>=1 with a(0)=1; i.e., a(n) equals 1/n times the coefficient of x^(n-1) in A(x)^(2n) for n>=1. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 2008]

(PARI) {a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[ #A]=-Vec(subst(Ser(A), x, x/Ser(A)^2))[ #A]); A[n+1]}

(PARI) /* This sequence is generated when k=2, m=0: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n, k=2, m=0)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 2008]

Cf. A120971; variants: A120972, A120974, A120976, A030266, A067145, A107096.

Cf. related variants: A145347, A145348, A147664, A145349, A145350. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 2008]

Sequence in context: A009636 A156272 A116364 this_sequence A111558 A001193 A161391

Adjacent sequences: A120967 A120968 A120969 this_sequence A120971 A120972 A120973

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 20 2006