OFFSET
1,2
COMMENTS
Conjecture: In general, if k > 0 and g.f. A(x) satisfies A((k+2)*x - (k+1)*A(x)) = x - (k*x)^2, then a(n) ~ (-1)^n * c(k) * (k*(k+1)/log(k+1))^n * n! / n^((k-1)/(k+1) + (k-2)*log(k+1)/k), where c(k) is a constant. - Vaclav Kotesovec, Oct 09 2017
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..300 (terms 1..200 from Paul D. Hanna)
FORMULA
a(n) ~ (-1)^n * c * 20^n * n! / (n^(3/5 + log(5)/2) * (log(5))^n), where c = 0.04310841216382017737... - Vaclav Kotesovec, Oct 09 2017
EXAMPLE
G.f.: A(x) = x + 4*x^2 - 40*x^3 + 1400*x^4 - 68000*x^5 + 4350000*x^6 - 335400000*x^7 + 30085500000*x^8 - 3064470000000*x^9 + 348742875000000*x^10 - 43821408750000000*x^11 + 6025223212500000000*x^12 - 899952237525000000000*x^13 + 145148307600812500000000*x^14 - 25148944415520000000000000*x^15 +...
such that A( 6*x - 5*A(x) ) = x - 16*x^2.
RELATED SERIES.
Define Ai(x) such that Ai(A(x)) = x, then Ai(x) begins:
Ai(x) = x - 4*x^2 + 72*x^3 - 2520*x^4 + 123424*x^5 - 7710448*x^6 + 579543872*x^7 - 50689868256*x^8 + 5046335530880*x^9 - 562904575158208*x^10 +...
where Ai(x - 16*x^2) = 6*x - 5*A(x).
PROG
(PARI) {a(n) = my(A=x, V=[1, 4]); for(i=1, n, V = concat(V, 0); A=x*Ser(V); V[#V] = Vec( subst(A, x, 6*x - 5*A) )[#V]/4 ); V[n]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Sep 23 2017
STATUS
approved