OFFSET
0,3
COMMENTS
Note that the g.f. is an even function, so only the coefficients of even powers of x are shown.
The g.f. A(x) of this sequence is motivated by the following identity:
Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) ;
here, p = x*A(x), q = -x*A(x), and r = 2*A(x), where A(x) causes the sum to equal unity.
FORMULA
The g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) satisfies the following relations.
(1) 1 = Sum_{n>=0} x^n*A(x)^n/(1 + 2^n*x*A(x)^(n+1)).
(2) 1 = Sum_{n>=0} (-x)^n*A(x)^n/(1 - 2^n*x*A(x)^(n+1)).
EXAMPLE
G.f.: A(x) = 1 + x^2 - 21*x^4 + 3260*x^6 - 1889209*x^8 + 4064654417*x^10 - 33841617523436*x^12 + 1114603532682365464*x^14 + ...
where
1 = 1/(1 + x*A(x)) + x*A(x)/(1 + 2*x*A(x)^2) + x^2*A(x)^2/(1 + 2^2*x*A(x)^3) + x^3*A(x)^3/(1 + 2^3*x*A(x)^4) + x^4*A(x)^4/(1 + 2^4*x*A(x)^5) + ...
also
1 = 1/(1 - x*A(x)) - x*A(x)/(1 - 2*x*A(x)^2) + x^2*A(x)^2/(1 - 2^2*x*A(x)^3) - x^3*A(x)^3/(1 - 2^3*x*A(x)^4) + x^4*A(x)^4/(1 - 2^4*x*A(x)^5) + ...
PROG
(PARI) {a(n) = my(A=1, V=[1]); for(i=0, 2*n, V=concat(V, 0); A = Ser(concat(V, 0)); V[#V] = polcoeff( sum(m=0, #V+1, x^m*A^m/(1 + 2^m*x*A^(m+1)) ), #V+1)/2 ); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A=1, V=[1]); for(i=0, 2*n, V=concat(V, 0); A = Ser(concat(V, 0)); V[#V] = polcoeff( sum(m=0, #V+1, (-x)^m*A^m/(1 - 2^m*x*A^(m+1)) ), #V+1)/2 ); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jan 08 2021
STATUS
approved