A388734 - OEIS

%I #7 Sep 21 2025 09:30:14

%S 1,1,3,9,33,127,517,2173,9379,41293,184765,837695,3840089,17768585,

%T 82879635,389283441,1839640281,8740574143,41728175821,200070896725,

%U 962982898195,4651303241605,22537977286117,109526693482111,533685200346481,2606863801375057,12762564311227107,62613992051698905

%N G.f. satisfies A(x) = 1 + x*A(x)^2 + x^2*(1-x)*A(x)^3.

%C All terms appear to be odd.

%H Paul D. Hanna, <a href="/A388734/b388734.txt">Table of n, a(n) for n = 0..730</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) A(x) = 1 + x*A(x)^2 + x^2*(1-x)*A(x)^3.

%F (2) 1 - x = Sum_{n>=0} (-1)^(n mod 3) * x^n * A(x)^n.

%F (3) A(x) = (1 - x^3*A(x)^3)/(1 - x*A(x) - x^2*A(x)^2).

%F (4) A(x) = Series_Reversion( x*(1 - x - x^2)/(1 - x^3) ).

%F (5) A( x*(1 - x - x^2)/(1 - x^3) ) = (1 - x^3)/(1 - x - x^2).

%e G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 33*x^4 + 127*x^5 + 517*x^6 + 2173*x^7 + 9379*x^8 + 41293*x^9 + 184765*x^10 + ...

%e where A(x) = 1 + x*A(x)^2 + x^2*(1-x)*A(x)^3.

%e Also,

%e 1 - x = 1 - x*A(x) + x^2*A(x)^2 + x^3*A(x)^3 - x^4*A(x)^4 + x^5*A(x)^5 + x^6*A(x)^6 - x^7*A(x)^7 + x^8*A(x)^8 +-+ ...

%e RELATED SERIES.

%e A(x)^2 = 1 + 2*x + 7*x^2 + 24*x^3 + 93*x^4 + 374*x^5 + 1567*x^6 + 6736*x^7 + 29581*x^8 + 132070*x^9 + 597755*x^10 + ...

%e A(x)^3 = 1 + 3*x + 12*x^2 + 46*x^3 + 189*x^4 + 795*x^5 + 3438*x^6 + 15150*x^7 + 67845*x^8 + 307785*x^9 + 1411638*x^10 + ...

%e Let F(x) satisfy A(x/F(x)) = F(x) then

%e F(x) = 1 + x + 2*x^2*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + ... + Fibonacci(n+1)*x^n + ...).

%o (PARI) {a(n) = my(A=1); A = (1/x)*serreverse( x*(1 - x - x^2)/(1 - x^3 +x*O(x^n)) ); polcoef(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A163733.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 20 2025