OFFSET
1,2
COMMENTS
Radius of convergence is r = (sqrt(41) - 5)/8, where r = r^2/(1 - 4*r - 4*r^2), with A(r) = 1.
Related identities:
(1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).
(2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).
More generally, if
F(x)^2 = F( x^2/(1 - 2*a*x + 2*(a^2 - b)*x^2) ),
then
F( x/(1 + a*x + b*x^2) )^2 = F( x^2/(1 + a^2*x^2 + b^2*x^4) );
here, a = 2, b = 6.
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 satisfies:
(1) A( x/(1 + 2*x + 6*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 6^2*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3.a) A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).
(3.b) A(x)^2 = -A( -x^2/(1 - 4*x - 8*x^2) ).
(4.a) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 8*x^2) ).
(4.b) A( x/(1 + 2*x) )^2 = -A( -x^2/(1 - 12*x^2) ).
(4.c) A( x/(1 + 2*x) )^2 = A( -x/(1 - 2*x) )^2.
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 9*x^3 + 38*x^4 + 176*x^5 + 832*x^6 + 4039*x^7 + 19938*x^8 + 99861*x^9 + 506042*x^10 + 2590099*x^11 + 13370898*x^12 + ...
where A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 112*x^5 + 585*x^6 + 3052*x^7 + 16018*x^8 + 84384*x^9 + 446384*x^10 + 2370240*x^11 + 12631104*x^12 + ...
(x*A(x))^(1/2) = x + x^2 + 4*x^3 + 15*x^4 + 65*x^5 + 291*x^6 + 1356*x^7 + 6474*x^8 + 31555*x^9 + 156315*x^10 + 784924*x^11 + ... + A357785(n)*x^n + ...
x/Series_Reversion(A(x)) = 1 + 2*x + 5*x^2 - 10*x^4 + 50*x^6 - 305*x^8 + 2025*x^10 - 14400*x^12 + 107500*x^14 - 829415*x^16 + 6559700*x^18 - 52908950*x^20 + ...
PROG
(PARI) {a(n) = my(A=x); for(i=1, #binary(n+1),
A = sqrt( subst(A, x, x^2/(1 - 4*x - 4*x^2 +x*O(x^n)) ) )
); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 01 2022
STATUS
approved