OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1001
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(3*x^2) = 2*x^2 + A(x)^2 - 2*A(x)^3.
(2) A(3*R(x)^2) = 2*R(x)^2 + x^2 - 2*x^3 where A(R(x)) = x.
(3.a) [x^(2*n+1)] A(x)^2 = [x^(2*n+1)] 2*A(x)^3 for n >= 1.
(3.b) [x^(2*n)] A(x)^2 = 3^n*a(n) + [x^(2*n)] 2*A(x)^3 for n >= 2.
(4.a) A(x)^3 == A(x^3) (mod 3) (conjecture).
(4.b) A(x)^2 == x^2 + 2*A(x^3) (mod 3) (conjecture).
EXAMPLE
G.f.: A(x) = x + x^2 + 7*x^3 + 17*x^4 + 147*x^5 + 445*x^6 + 2199*x^7 + 9993*x^8 + 61837*x^9 + 281417*x^10 + 1515387*x^11 + 7844861*x^12 + ...
where A(3*x^2) = 2*x^2 + A(x)^2 - 2*A(x)^3.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 15*x^4 + 48*x^5 + 377*x^6 + 1422*x^7 + 7635*x^8 + 35612*x^9 + 211185*x^10 + 1032006*x^11 + 5643619*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 24*x^5 + 94*x^6 + 711*x^7 + 3129*x^8 + 17806*x^9 + 87732*x^10 + 516003*x^11 + 2659607*x^12 + ...
A(x)^2 - 2*A(x)^3 = x^2 + 9*x^4 + 189*x^6 + 1377*x^8 + 35721*x^10 + 324405*x^12 + 4809213*x^14 + ... + 3^n*a(n)*x^(2*n) + ...
PROG
(PARI) {a(n) = my(V=[0, 1], A=x); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef(( subst(A, x, 3*x^2) - 2*x^2 - A^2 + 2*A^3 )/2, #V) ); V[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 26 2025
STATUS
approved