A350474 - OEIS

%I #10 Sep 01 2022 12:01:09

%S 1,1,4,17,89,487,2835,17039,105390,665364,4273038,27820758,183232742,

%T 1218574419,8171805764,55196973265,375190697191,2564513095021,

%U 17615896411327,121541577797136,841923706799550,5853065006349460,40823876909916688

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

%e G.f.: A(x) = x + x^3 + 4*x^5 + 17*x^7 + 89*x^9 + 487*x^11 + 2835*x^13 + 17039*x^15 + 105390*x^17 + 665364*x^19 + ...

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

%e R(x) = x - x^3 - x^5 + 3*x^7 - 4*x^9 + 6*x^11 - 7*x^13 - 11*x^15 + 54*x^17 - 68*x^19 + ... + A350475(n)*x^(2*n-1) + ...

%e and which by definition also satisfies A(x*R(x)) = x^2 - x^4.

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

%e F(0) = x,

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

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

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

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

%e F(5) = (1 - x^32 * (1-x^2)^16 * F(2)^8 * F(3)^4 * F(4)^2),

%e ...

%e F(n+1) = 1 - (1 - F(n))^2 * 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^4*(1-x^2)^2) * (1 - x^8*(1-x^2)^4*(1 - x^2*(1-x^2)^2)^2) * (1 - x^16*(1-x^2)^8*(1 - x^2*(1-x^2)^2)^4*(1 - x^4*(1-x^2)^4*(1 - x^2*(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^2*(1 - x^2) - subst(x*Ser(A),x, x * serreverse(x*Ser(A))), #A+1) );A[n]}

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

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

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

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

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

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

%Y Cf. A350475 (inverse), A273162, A273203, A350434, A350476, A350478, A350480, A350482.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jan 01 2022