OFFSET
1,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
If A(B(x)) = x, then g.f. A(x) and B(x) satisfy:
(1) x*A(x) - A(x)^2 = B(-x^3).
(2) A(x) = x - x*C( B(-x^3)/x^2 ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
a(n) ~ (-1)^n * c * d^n / n^(3/2), where d = 3.9374997379511376037..., c = 0.034997955229443779... . - Vaclav Kotesovec, Jun 24 2016
EXAMPLE
G.f.: A(x) = x + x^2 - x^3 + 2*x^4 - 4*x^5 + 12*x^6 - 36*x^7 + 115*x^8 - 366*x^9 + 1202*x^10 - 4016*x^11 + 13684*x^12 - 47192*x^13 + 164550*x^14 +...
such that A( x*A(x) - A(x)^2 ) = -x^3.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 - x^4 + 2*x^5 - 3*x^6 + 12*x^7 - 36*x^8 + 118*x^9 - 366*x^10 + 1202*x^11 - 4004*x^12 + 13684*x^13 - 47192*x^14 + 164604*x^15 +...
x*A(x) - A(x)^2 = -x^3 - x^6 - 3*x^9 - 12*x^12 - 54*x^15 - 264*x^18 - 1362*x^21 - 7300*x^24 - 40245*x^27 - 226746*x^30 - 1299779*x^33 +...
Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then
B(x) = x - x^2 + 3*x^3 - 12*x^4 + 54*x^5 - 264*x^6 + 1362*x^7 - 7300*x^8 + 40245*x^9 - 226746*x^10 + 1299779*x^11 - 7556310*x^12 + 44445150*x^13 - 264010326*x^14 + 1581537357*x^15 - 9543458802*x^16 + 57956158488*x^17 - 353941849554*x^18 +...
such that x*A(x) - A(x)^2 = B(-x^3).
PROG
(PARI) {a(n) = my(A=[1, 1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = polcoeff(x^3 + subst(F, x, x*F - F^2), #A+1) ); A[n]}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 12 2016
STATUS
approved