OFFSET
1,2
COMMENTS
Conjecture: a(n) is odd iff n = 4^k for k >= 0.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..400
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas, in which C(x) = x + C(x)^2 is the Catalan function (A000108).
(1) x = A(x - x^2) - A(x + x^2)^2.
(2) A(x) = C(x) + A(2*C(x) - x)^2.
(3) A(x)^2 = C(-x) + A(-2*C(-x) - x).
(4) A(x) = A( 1-x - sqrt(1-4*x) )^2 + (1 - sqrt(1-4*x))/2.
(5) A(x)^2 = A( sqrt(1+4*x) - 1-x ) + (1 - sqrt(1+4*x))/2.
EXAMPLE
G.f. A(x) = x + 2*x^2 + 10*x^3 + 65*x^4 + 508*x^5 + 4534*x^6 + 44836*x^7 + 482214*x^8 + 5571102*x^9 + 68535590*x^10 + ...
where x = A(x - x^2) - A(x + x^2)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 24*x^4 + 170*x^5 + 1376*x^6 + 12400*x^7 + 122193*x^8 + 1300492*x^9 + 14815264*x^10 + ...
where A(x)^2 = C(-x) + A(-2*C(-x) - x) and C(x) is the Catalan function.
A(x-x^2) = x + x^2 + 6*x^3 + 37*x^4 + 278*x^5 + 2374*x^6 + 22452*x^7 + 231357*x^8 + 2566806*x^9 + 30395906*x^10 + ...
A(x+x^2) = x + 3*x^2 + 14*x^3 + 97*x^4 + 798*x^5 + 7474*x^6 + 77380*x^7 + 869221*x^8 + 10463590*x^9 + 133815278*x^10 + ...
where A(x+x^2)^2 = A(x-x^2) - x.
PROG
(PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( x - subst(Ax, x, x - x^2 ) + subst(Ax, x, x + x^2 )^2, #A-1) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 14 2024
STATUS
approved