A350482 - OEIS

%I #7 Sep 01 2022 11:59:43

%S 1,1,3,12,56,282,1494,8207,46332,267174,1566994,9318630,56058288,

%T 340530734,2085902781,12869679276,79906734738,498903972318,

%U 3130391261901,19728829226876,124833794989761,792731281119894,5050567538514688,32273851864657050,206799487942332132

%N G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) satisfies: A(x^3*R(x)) = x^4 - x^6, 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^2, and F(n+1) = 1 - (1 - F(n))^4 * F(n)^2 for n > 0.

%e G.f.: A(x) = x + x^3 + 3*x^5 + 12*x^7 + 56*x^9 + 282*x^11 + 1494*x^13 + 8207*x^15 + 46332*x^17 + 267174*x^19 + ...

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

%e R(x) = x - x^3 - x^9 + 3*x^11 - 3*x^13 + x^15 - x^33 + 9*x^35 - 36*x^37 + 84*x^39 + ... + A350483(n)*x^(2*n-1) + ...

%e and which by definition also satisfies A(x^3*R(x)) = x^4 - x^6.

%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 2*6^(n-1), by the following recurrence:

%e F(0) = x,

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

%e F(2) = (1 - x^8 * (1-x^2)^2),

%e F(3) = (1 - x^32 * (1-x^2)^8 * F(2)^2),

%e F(4) = (1 - x^128 * (1-x^2)^32 * F(2)^8 * F(3)^2),

%e F(5) = (1 - x^512 * (1-x^2)^128 * F(2)^32 * F(3)^8 * F(4)^2),

%e ...

%e F(n+1) = 1 - (1 - F(n))^4 * F(n)^2

%e ...

%e Then the series reversion R(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^2) * (1 - x^8*(1-x^2)^2) * (1 - x^32*(1-x^2)^8*(1 - x^8*(1-x^2)^2)^2) * (1 - x^128*(1-x^2)^32*(1 - x^8*(1-x^2)^2)^8*(1 - x^32*(1-x^2)^8*(1 - x^8*(1-x^2)^2)^2)^2) * ...

%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^4*(1 - x^2) - subst(x*Ser(A),x, x^3 * serreverse(x*Ser(A))), #A+3) );A[n]}

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

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

%o N = 300; \\ set limit on order of polynomials to be 2 times desired number of terms

%o {F(n) = my(G=x); if(n==0,G=x, if(n==1, G = (1-x^2), G = 1 - (1 - F(n-1))^4 * F(n-1)^2 +x*O(x^N) ));G}

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

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

%Y Cf. A350483 (inverse), A273162, A273203, A350434, A350474, A350476, A350478, A350480.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jan 01 2022