A350432 - OEIS

%I #13 Jan 26 2024 10:03:04

%S 1,1,1,2,2,3,5,9,14,18,25,38,60,97,159,265,444,735,1187,1865,2851,

%T 4271,6378,9621,14724,22864,35947,57044,91141,146384,236102,382124,

%U 620298,1009685,1647703,2694709,4413524,7232548,11845740,19369888,31590755,51346902,83126317

%N G.f. A(x) satisfies: A(x) = A(x^2 + x^3) / x.

%C The radius of convergence r of the g.f. A(x) is r = (sqrt(5) - 1)/2. - _Paul D. Hanna_, Jan 26 2024

%H Paul D. Hanna, <a href="/A350432/b350432.txt">Table of n, a(n) for n = 1..2050</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

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

%F (2) R(x*A(x)) = x^2 + x^3, where R(A(x)) = x.

%F (3) A(x) = x * Product_{n>=1} F(n), where F(1) = 1+x, and F(n+1) = 1 + (F(n) - 1)^2 * F(n) for n >= 1.

%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 5*x^7 + 9*x^8 + 14*x^9 + 18*x^10 + 25*x^11 + 38*x^12 + 60*x^13 + ...

%e Let R(x) be the series reversion of A(x),

%e R(x) = x - x^2 + x^3 - 2*x^4 + 6*x^5 - 17*x^6 + 45*x^7 - 123*x^8 + 360*x^9 - 1085*x^10 + 3271*x^11 - 9905*x^12 + 30417*x^13 + ... + A350431(n)*x^n + ...

%e then R(x) and g.f. A(x) satisfy:

%e (1) R(A(x)) = x,

%e (2) R(x*A(x)) = x^2 + x^3.

%e GENERATING METHOD.

%e Define F(n), a polynomial in x of order 3^(n-1), by the following recurrence:

%e F(1) = (1 + x),

%e F(2) = (1 + x^2 * (1+x)),

%e F(3) = (1 + x^4 * (1+x)^2 * F(2)),

%e F(4) = (1 + x^8 * (1+x)^4 * F(2)^2 * F(3)),

%e F(5) = (1 + x^16 * (1+x)^8 * F(2)^4 * F(3)^2 * F(4)),

%e ...

%e F(n+1) = 1 + (F(n) - 1)^2 * F(n)

%e ...

%e Then the g.f. A(x) equals the infinite product:

%e A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...

%e that is,

%e A(x) = x * (1+x) * (1 + x^2*(1+x)) * (1 + x^4*(1+x)^2*(1 + x^2*(1+x))) * (1 + x^8*(1+x)^4*(1 + x^2*(1+x))^2*(1 + x^4*(1+x)^2*(1 + x^2*(1+x)))) * ...

%o (PARI) /* Using Functional Equation in Definition */

%o {a(n) = my(A=x); for(i=0,#binary(n),

%o A = subst(A,x, x^2*(1 + x) +x^2*O(x^n) )/x ); polcoeff(A,n)}

%o for(n=1,50,print1(a(n),", "))

%o (PARI) /* Using Infinite Product Formula */

%o {F(n) = my(G=x); if(n==0,G=x, if(n==1,G=1+x, G = 1 + (F(n-1) - 1)^2 * F(n-1) ));G}

%o {a(n) = my(A = prod(k=0,#binary(n), F(k) +x*O(x^n))); polcoeff(A,n)}

%o for(n=1,50,print1(a(n),", "))

%Y Cf. A350431 (inverse), A273162, A350433, A350434.

%Y Cf. A369545, A369546, A369547, A369548, A369549.

%Y Cf. A369552, A369553, A369554, A369555, A369556.

%K nonn

%O 1,4

%A _Paul D. Hanna_, Dec 30 2021