OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..500
FORMULA
a(n) ~ c * d^n / n^(3/2), where d = 6.426300389361325672327464898259648... and c = 0.035189970759375828095135204598637... . - Vaclav Kotesovec, May 03 2016
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 9*x^4 + 40*x^5 + 192*x^6 + 959*x^7 + 4988*x^8 + 26587*x^9 + 144672*x^10 + 800229*x^11 + 4486914*x^12 +...
where A( x*A(x) - A(x)^3 ) = x^2.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 40*x^6 + 192*x^7 + 963*x^8 + 4988*x^9 + 26589*x^10 + 144672*x^11 + 800253*x^12 + 4486914*x^13 + 25444374*x^14 +...
x*A(x) - A(x)^3 = x^2 - x^4 - 4*x^8 - 2*x^10 - 24*x^12 - 19*x^14 - 206*x^16 - 194*x^18 - 1980*x^20 - 2390*x^22 - 20920*x^24 - 31626*x^26 - 236114*x^28 +...
Let B(x) be the series reversion of the g.f. A(x), so that A(B(x)) = x, then
B(x) = x - x^2 - 4*x^4 - 2*x^5 - 24*x^6 - 19*x^7 - 206*x^8 - 194*x^9 - 1980*x^10 - 2390*x^11 - 20920*x^12 - 31626*x^13 - 236114*x^14 +...
such that B(x^2) = x*A(x) - A(x)^3.
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -Vec(subst(F, x, x*F - F^3))[#A] ); A[n]}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 27 2016
STATUS
approved