A240999 G.f. satisfies: A(x)^2 = x + A(x*A(x)^3).

1, 1, 2, 17, 233, 4363, 101905, 2831645, 90903590, 3305297011, 134176657131, 6014350289121, 295061981283195, 15728427054231126, 905372551016907180, 55980148124997746680, 3700788877216739351042, 260516910027328463667728, 19457399278222418766271255

Offset

0,3

Comments

In general, if g.f. satisfies: A(x)^2 = x + A(x*A(x)^q), q > 1, then a(n) ~ c(q) * q^n * n^(n - 1/q + (1/2 - 3/(2*q))*log(2)) / (exp(n) * log(2)^n), where c(q) is a constant independent on n.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

a(n) ~ c * 3^n * n^(n-1/3) / (exp(n) * (log(2))^n), where c = 0.3223127444389467551929... . - Vaclav Kotesovec, Aug 08 2014

Example

G.f.: A(x) = 1 + x + 2*x^2 + 17*x^3 + 233*x^4 + 4363*x^5 + 101905*x^6 +...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 38*x^3 + 504*x^4 + 9260*x^5 + 213757*x^6 +...
A(x*A(x)^3) = 1 + x + 5*x^2 + 38*x^3 + 504*x^4 + 9260*x^5 + 213757*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 64*x^3 + 819*x^4 + 14754*x^5 + 336467*x^6 +...

Prog

(PARI) {a(n)=local(A=[1, 1], Ax); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A]=Vec(1+subst(Ax, x, x*Ax^3) - Ax^2)[#A]); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A240996 (q=2), A241996 (q=4), A241997 (q=5), A241998 (q=6), A241999 (q=7).

Sequence in context: A006227 A307289 A036082 * A319947 A361194 A342205

Adjacent sequences: A240996 A240997 A240998 * A241000 A241001 A241002

Keyword

nonn

Author

Paul D. Hanna, Aug 06 2014

Status

approved