login
G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) satisfies: A(x*R(x)) = x^2 - x^5, where A(R(x)) = x.
5

%I #7 Sep 01 2022 12:00:49

%S 1,1,5,29,203,1519,12047,99112,839142,7260953,63934050,570993752,

%T 5160035536,47097134677,433541911687,4020342022925,37521976803972,

%U 352181765052016,3322233412518174,31480621945734200,299508608455316864,2859957234218787021,27399875112420058167

%N G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) satisfies: A(x*R(x)) = x^2 - x^5, where A(R(x)) = x.

%F G.f.: A(x) = Series_Reversion( Product_{n>=0} F(n) ), where F(0) = x, F(1) = 1-x^3, and F(n+1) = 1 - (1 - F(n))^2 * F(n)^3 for n > 0.

%e G.f.: A(x) = x + x^4 + 5*x^7 + 29*x^10 + 203*x^13 + 1519*x^16 + 12047*x^19 + 99112*x^22 + 839142*x^25 + ...

%e The series reversion is here denoted R(x) so that R(A(x)) = x where

%e R(x) = x - x^4 - x^7 + 4*x^10 - 7*x^13 + 11*x^16 - 18*x^19 - 5*x^22 + 138*x^25 - 368*x^28 + ... + A350477(n)*x^(3*n-2) + ...

%e and which by definition also satisfies A(x*R(x)) = x^2 - x^5.

%e GENERATING METHOD.

%e One may obtain the g.f. A(x) from the following method used to generate the series reversion R(x).

%e Define F(n), a polynomial in x of order 3*5^(n-1), by the following recurrence:

%e F(0) = x,

%e F(1) = (1 - x^3),

%e F(2) = (1 - x^6 * (1-x^3)^3),

%e F(3) = (1 - x^12 * (1-x^3)^6 * F(2)^3),

%e F(4) = (1 - x^24 * (1-x^3)^12 * F(2)^6 * F(3)^3),

%e F(5) = (1 - x^48 * (1-x^3)^24 * F(2)^12 * F(3)^6 * F(4)^3),

%e ...

%e F(n+1) = 1 - (1 - F(n))^2 * F(n)^3

%e ...

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

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

%e that is,

%e R(x) = x * (1-x^3) * (1 - x^6*(1-x^3)^3) * (1 - x^12*(1-x^3)^6*(1 - x^6*(1-x^3)^3)^3) * (1 - x^24*(1-x^3)^12*(1 - x^6*(1-x^3)^3)^6*(1 - x^12*(1-x^3)^6*(1 - x^6*(1-x^3)^3)^3)^3) * ...

%e The g.f. of this sequence is then obtained as the series reversion of this infinite product.

%o (PARI) {a(n) = my(A=[1,0]); for(i=1,n, A=concat(A,0);

%o A[#A] = -polcoeff( x^2*(1 - x^3) - subst(x*Ser(A),x, x * serreverse(x*Ser(A))), #A+1) );A[n]}

%o for(n=1,30,print1(a(3*n-2),", "))

%o (PARI) /* Using Infinite Product Formula for Series Reversion */

%o N = 300; \\ set limit on order of polynomials to be 3 times desired number of terms

%o {F(n) = my(G=x); if(n==0,G=x, if(n==1, G = (1-x^3), G = 1 - (1 - F(n-1))^2 * F(n-1)^3 +x^2*O(x^N) ));G}

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

%o A = serreverse(B); polcoeff(H=A,n)}

%o for(n=1,30,print1(a(3*n-2),", "))

%Y Cf. A350477 (inverse), A273162, A273203, A350434, A350474, A350478, A350480, A350482.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jan 01 2022