OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..350
FORMULA
G.f. satisfies:
(1) A(x) = 1 + 2*Series_Reversion( x*(1 - 3*x - 2*x^2)/(1 + 2*x) ).
(2) A(x) = Sum_{n>=0} C(3*n,n)/(2*n+1) * (4 + 4*x*A(x))^(2*n+1) / 5^(3*n+1).
(3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) = (3+x + G(x)^3)/4 is the g.f. of A120592.
Recurrence: 17*(n-1)*n*a(n) = 150*(n-1)*(2*n-3)*a(n-1) - 40*(6*n^2 - 24*n + 23)*a(n-2) + 32*(n-2)*(2*n-7)*a(n-3). - Vaclav Kotesovec, Jul 03 2014
a(n) ~ sqrt(15-9*2^(2/3)) * ((60*2^(2/3)+72*2^(1/3)+100)/17)^n / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 03 2014
EXAMPLE
G.f.: A(x) = 1 + 2*x + 10*x^2 + 84*x^3 + 882*x^4 + 10380*x^5 + 130916*x^6 +...
Compare
(5 - 4*x)*A(x) = 5 + 6*x + 42*x^2 + 380*x^3 + 4074*x^4 + 48372*x^5 +...
to:
A(x)^3 = 1 + 6*x + 42*x^2 + 380*x^3 + 4074*x^4 + 48372*x^5 + 613060*x^6 +...
PROG
(PARI) {a(n)=polcoeff(1 + 2*serreverse(x*(1 - 3*x - 2*x^2)/(1 + 2*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=[1], Ax=1+x); for(i=1, n, A=concat(A, 0); Ax=Ser(A); A[#A]=Vec( ( Ax^3 - (5-4*x)*Ax )/2 )[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 02 2014
STATUS
approved