A249933 G.f. A(x) satisfies: 3*A(x) - 2*(1+x) = Series_Reversion( x/(2*A(x) - (1+x)) )/x.

1, 1, 1, 7, 77, 1111, 19389, 391959, 8942541, 226359559, 6281444909, 189409746967, 6163853164333, 215283653614823, 8033015072981677, 318963167792376087, 13430607648848415309, 597871639955925563463, 28059143187405536116653, 1384849551267166030082199, 71712434188939188553335789

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Formula

G.f. A(x) satisfies: 2*A(x) - (1+x) = x/Series_Reversion( x*(3*A(x) - 2*(1+x)) ).

G.f. A(x) satisfies: 2*A(3*x*A(x) - 2*x*(1+x)) = 3*(1+x)*A(x) - 2*(1+x)^2 + 1.

Define B(x) = 2*A(x) - (1+x) and C(x) = 3*A(x) - 2*(1+x), then B(x) = C(x/B(x)) and C(x) = B(x*C(x)).

a(n) ~ c * n^(n + 1/2 + 2*log(3/2)) / (exp(n) * (log(3/2))^n), where c = 0.104713270868379... . - Vaclav Kotesovec, Nov 30 2014

Example

G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 77*x^4 + 1111*x^5 + 19389*x^6 +...
If we define
B(x) = 2*A(x) - (1+x) = 1 + x + 2*x^2 + 14*x^3 + 154*x^4 + 2222*x^5 +...
C(x) = 3*A(x) - 2*(1+x) = 1 + x + 3*x^2 + 21*x^3 + 231*x^4 + 3333*x^5 +...
then
B(x) = C(x/B(x)) and
C(x) = B(x*C(x)).
Further, A(x), B(x), and C(x) satisfy
B(x) = 3*A(x/B(x)) - 2 - 2*x/B(x) and
C(x) = (2*A(x*C(x)) - 1)/(1+x).

Prog

(PARI) /* 3*A(x) - 2*(1+x) = Series_Reversion(x/(2*A(x) - (1+x)))/x */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=Ser(concat(Vec(A), 0));
A = serreverse(x/(2*A - (1+x)))/x + 2*(1+x) - 2*A); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* 2*A(x) - (1+x) = x/Series_Reversion(x*(3*A(x) - 2*(1+x))) */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=Ser(concat(Vec(A), 0));
A = 3*A - x/serreverse(x*(3*A - 2*(1+x))) - (1+x)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A120955.

Sequence in context: A267709 A234466 A306031 * A107866 A034176 A001765

Adjacent sequences: A249930 A249931 A249932 * A249934 A249935 A249936

Keyword

nonn

Author

Paul D. Hanna, Nov 26 2014

Status

approved