OFFSET
1,2
COMMENTS
Radius of convergence is r = 1/9 where r = r^2/(1-6*r)^2 with A(r) = 1.
Compare to: C(x)^2 = C( x^2/(1-2*x)^2 ) where C(x) = (1-2*x-sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers A000108.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..520
FORMULA
G.f. satisfies:
(1) A(x) = -A( -x/(1-12*x) ).
(2) A(x^2) = A( x/(1+6*x) )^2 = A( -x/(1-6*x) )^2.
(3) A( x/(1+3*x)^2 ) = -A( -x/(1-3*x)^2 ), an odd function.
(4) A( x/(1+3*x)^2 )^2 = A( x^2/(1+9*x^2)^2 ), an even function.
(5) A( x/(1+4*x) ) = G(x) = Sum_{n>=1} A264224(n)*x^n where G(x)^2 = G( x^2/(1-4*x) ).
(6) A( x/(1+8*x) ) = -G(-x) = Sum_{n>=1} (-1)^(n-1) * A264224(n)*x^n where G(x)^2 = G( x^2/(1-4*x) ).
Sum_{k=0..n} binomial(n,k) * (-6)^(n-k) * a(k+1) = 0 for odd n.
Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * a(k+1) = A264224(n+1) for n>=0.
Sum_{k=0..n} binomial(n,k) * (-8)^(n-k) * a(k+1) = (-1)^n * A264224(n+1) for n>=0.
EXAMPLE
G.f.: A(x) = x + 6*x^2 + 39*x^3 + 270*x^4 + 1959*x^5 + 14706*x^6 + 113166*x^7 + 887004*x^8 + 7050837*x^9 + 56672622*x^10 + 459646488*x^11 + 3756181248*x^12 +...
where A( x^2/(1-6*x)^2 ) = A(x)^2,
A( x^2/(1-6*x)^2 ) = x^2 + 12*x^3 + 114*x^4 + 1008*x^5 + 8679*x^6 + 73980*x^7 + 628506*x^8 + 5336928*x^9 + 45351591*x^10 + 385869348*x^11 + 3287962710*x^12 +...
Also, A( x/(1+6*x) ) = A(x^2)^(1/2),
A( x/(1+6*x) ) = x + 3*x^3 + 15*x^5 + 90*x^7 + 597*x^9 + 4212*x^11 + 30942*x^13 + 233766*x^15 + 1802706*x^17 + 14122359*x^19 + 112033791*x^21 + 898024320*x^23 +...
Let B(x) = x/Series_Reversion( A(x) ), so that A(x) = x*B(A(x)), then
B(x) = 1 + 6*x + 3*x^2 - 3*x^4 + 9*x^6 - 33*x^8 + 126*x^10 - 513*x^12 + 2214*x^14 - 9876*x^16 + 45045*x^18 - 209493*x^20 + 990198*x^22 +...+ A264412(n)*x^(2*n) +...
such that B(x) = F(x^2) + 6*x = F(x)^2 where F(x) is the g.f. of A264412.
PROG
(PARI) {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-6*x +x*O(x^n))^2) ) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 16 2015
STATUS
approved