OFFSET
1,2
COMMENTS
Compare to F( F(x)^2/(1 + 2*F(x))^2 ) = x*F(x) when F(x) = x/(1 - 4*x).
Conjectures:
(1) a(n) == 0 (mod 3) for n > 0.
(2) a(4^n+1) == 6 (mod 9) and a(2*4^n+1) == 3 (mod 9) for n >= 0.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..520
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n along with its series reversion R(x), satisfies the following formulas.
(1) A( A(x)^2/(1 + 3*A(x))^2 ) = x*A(x).
(2) A( (1/x) * A(x^2/(1 + 3*x)^2) ) = x.
(3) A( x^2/(1 + 3*x)^2 ) = x*R(x) where A(R(x)) = x.
(4) A(x^2) = R( x/(1 - 3*x) ) * x/(1 - 3*x) where R(A(x)) = x.
(5) R( x*R(x) ) = x^2/(1 + 3*x)^2 where R(A(x)) = x.
EXAMPLE
G.f.: A(x) = x + 6*x^2 + 39*x^3 + 270*x^4 + 1959*x^5 + 14724*x^6 + 113706*x^7 + 896994*x^8 + 7198257*x^9 + 58580766*x^10 + ...
where A( A(x)^2/(1 + 3*A(x))^2 ) = x*A(x).
RELATED SERIES.
A(x)/(1 + 3*A(x)) = x + 3*x^2 + 12*x^3 + 63*x^4 + 393*x^5 + 2700*x^6 + 19638*x^7 + 148311*x^8 + 1150959*x^9 + 9120015*x^10 + ...
A(x)^2/(1 + 3*A(x))^2 = x^2 + 6*x^3 + 33*x^4 + 198*x^5 + 1308*x^6 + 9270*x^7 + 68877*x^8 + 528768*x^9 + 4157745*x^10 + ...
Let R(x) be the series reversion of A(x), R(A(x)) = x, then
R(x) = x - 6*x^2 + 33*x^3 - 180*x^4 + 984*x^5 - 5400*x^6 + 29754*x^7 - 164592*x^8 + 913938*x^9 - 5093244*x^10 + ...
SPECIFIC VALUES.
A(t) = 1 at t = A(1/16) = 0.10317279669579230295027576579252717989833579024...
A(t) = 1/2 at t = 2*A(1/25) = 0.1061508484665129500171873459941969232301870...
A(t) = 1/3 at t = 3*A(1/36) = 0.1003352603928806223818500963742672318320239...
A(t) = 1/4 at t = 4*A(1/49) = 0.0931746054665880225723638980700677838877663...
A(t) = 1/5 at t = 5*A(1/64) = 0.0862838920959754114526744600000633965142439...
A(t) = 1/6 at t = 6*A(1/81) = 0.0800427051964615614631388794610266659084609...
A(t) = 1/7 at t = 7*A(1/100) = 0.074493383010172543991350429378892726907769...
A(1/10) = 0.328082324724629985490856998205652119340000639...
A(1/11) = 0.231471123507693119766989879518138044746004716...
A(1/12) = 0.183189828931917973681273703531570036478268877...
A(1/13) = 0.152737756861086935804497387625417005550146684...
A(1/14) = 0.131415473598481160745300630661369723367840939...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A) - subst(Ser(A), x, Ser(A)^2/(1 + 3*Ser(A))^2 ), #A); ); A[n+1]}
for(n=1, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 10 2024
STATUS
approved