A325157 G.f.: A(x) = Sum_{n>=0} x^n * (1 + (-1)^n * A(x))^n / (1 - (-1)^n * x*A(x))^(n+1).

1, 1, 5, 21, 109, 573, 3209, 18409, 108569, 651897, 3977677, 24578237, 153522117, 967698037, 6147885585, 39325620689, 253063546417, 1637141058033, 10641294559637, 69461189473381, 455143665718301, 2992662818159277, 19739469132843673, 130576261281640377, 866046528206946377, 5758054698392120425, 38369626772586994013, 256215306174255250701, 1714211196437154993013, 11489680147035224255717, 77140806921880072224801

Offset

0,3

Comments

Compare the g.f. to the following related identities.

(1.a) F(x) = Sum_{n>=0} x^n * (1 + F(x))^n / (1 + x*F(x))^(n+1) holds when G(x) = 1/(1-x).

(1.b) F(x) = Sum_{n>=0} x^n * (1 - F(x))^n / (1 - x*F(x))^(n+1) holds when G(x) = 1/(1-x).

(2.a) G(x) = Sum_{n>=0} x^n * (1 + (-1)^n/G(x))^n / (1 - (-1)^n*x/G(x))^(n+1) holds when G(x) = (1 + sqrt(1 + 16*x^2))/(2*(1 - x)).

(2.b) G(x) = Sum_{n>=0} x^n * (1 - (-1)^n/G(x))^n / (1 + (-1)^n*x/G(x))^(n+1) holds when G(x) = (1 + sqrt(1 + 16*x^2))/(2*(1 - x)).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} x^n * (1 + (-1)^n * A(x))^n / (1 - (-1)^n * x*A(x))^(n+1).

(2) A(x) = Sum_{n>=0} x^n * (1 - (-1)^n * A(x))^n / (1 + (-1)^n * x*A(x))^(n+1).

(3) 0 = 4*x^2*A(x)^3 - (1-x)^2*A(x) + (1-x).

FORMULAS INVOLVING TERMS.

a(n) = 1 (mod 4) for n >= 0.

Example

G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 109*x^4 + 573*x^5 + 3209*x^6 + 18409*x^7 + 108569*x^8 + 651897*x^9 + 3977677*x^10 + 24578237*x^11 + 153522117*x^12 + ...
such that
A(x) = 1/(1 + x*A(x)) + x*(1 + A(x))/(1 - x*A(x))^2 + x^2*(1 - A(x))^2/(1 + x*A(x))^3 + x^3*(1 + A(x))^3/(1 - x*A(x))^4 + x^4*(1 - A(x))^4/(1 + x*A(x))^5 + x^5*(1 + A(x))^5/(1 - x*A(x))^6 + x^6*(1 - A(x))^6/(1 + x*A(x))^7 + ...
also,
A(x) = 1/(1 - x*A(x)) + x*(1 - A(x))/(1 + x*A(x))^2 + x^2*(1 + A(x))^2/(1 - x*A(x))^3 + x^3*(1 - A(x))^3/(1 + x*A(x))^4 + x^4*(1 + A(x))^4/(1 - x*A(x))^5 + x^5*(1 - A(x))^5/(1 + x*A(x))^6 + x^6*(1 + A(x))^6/(1 - x*A(x))^7 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=Vec(sum(n=0, #A, x^n*(1 + (-1)^n*Ser(A))^n/(1 - (-1)^n*x*Ser(A))^(n+1)))); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=Vec(sum(n=0, #A, x^n*(1 - (-1)^n*Ser(A))^n/(1 + (-1)^n*x*Ser(A))^(n+1)))); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A325156, A307123.

Sequence in context: A337168 A341853 A260845 * A218299 A121881 A041321

Adjacent sequences: A325154 A325155 A325156 * A325158 A325159 A325160

Keyword

nonn

Author

Paul D. Hanna, Apr 06 2019

Status

approved