OFFSET
0,2
FORMULA
From Paul D. Hanna, Apr 04 2022: (Start)
a(2*n+1) = 6 * 24^n * binomial(3*n+1,n)/(3*n+1) for n >= 0.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) (A(x)^2 + A(-x)^2)/2 = 1.
(2) (A(x)^3 - A(-x)^3)/2 = 18*x.
(3) (A(x) - A(-x))/2 = 6*Series_Reversion(x - 24*x^3).
(4) (A(x) + A(-x))/2 = sqrt( (1 + A(x)*A(-x))/2 ).
(5) A(x)*A(-x) = 1 - 72*Series_Reversion(x - 24*x^3)^2.
(6) A(x) = B(x) + sqrt(1 - B(x)^2), where B(x) = 6*Series_Reversion(x - 24*x^3).
(End)
EXAMPLE
G.f.: A(x) = 1 + 6*x - 18*x^2 + 144*x^3 - 1026*x^4 + 10368*x^5 +...
where
A(x)^2 = 1 + 12*x + 72*x^3 + 3240*x^5 + 229392*x^7 + 20083464*x^9 +...
A(x)^3 = 1 + 18*x + 54*x^2 + 1134*x^4 + 63180*x^6 + 4903254*x^8 +...
PROG
(PARI) {a(n)=local(A=[1, 6]); for(k=2, n, A=concat(A, 0); if(k%2==0, A[#A]=-Vec(Ser(A)^2)[#A]/2, A[#A]=-Vec(Ser(A)^3)[#A]/3)); A[n+1]}
(PARI) /* Using series reversion: */
{a(n) = my(A, B = 6*serreverse(x - 24*x^3 +x^2*O(x^n)) );
A = B + sqrt(1 - B^2); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Apr 04 2022
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Oct 06 2011
STATUS
approved