A307123 G.f.: A(x) = Sum_{n>=0} x^n * (2^n + A(x))^n / (1 + 2^n*x*A(x))^(n+1).

1, 2, 14, 456, 62242, 32889024, 68215273298, 561507179207912, 18430985645896044938, 2417187641870733621223872, 1267541802406230825844657147330, 2658386272037449312963901235020431528, 22300569741484633201700369303589604687459554, 748287098858623530181904979813004104177622808814080, 100433559671942995118342692594965847000464726608666998745586

Offset

0,2

Comments

Note that G(x) = Sum_{n>=0} x^n * (q^n + G(x))^n / (1 + q^n*x*G(x))^(n+1) holds at q = 1 when G(x) = 1/(1-x).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = 1 + 2*x + 14*x^2 + 456*x^3 + 62242*x^4 + 32889024*x^5 + 68215273298*x^6 + 561507179207912*x^7 + 18430985645896044938*x^8 + ...
such that
A(x) = 1/(1 + x*A(x)) + x*(2 + A(x))/(1 + 2*x*A(x))^2 + x^2*(2^2 + A(x))^2/(1 + 2^2*x*A(x))^3 + x^3*(2^3 + A(x))^3/(1 + 2^3*x*A(x))^3 + x^4*(2^4 + A(x))^4/(1 + 2^4*x*A(x))^4 + x^5*(2^5 + A(x))^5/(1 + 2^5*x*A(x))^6 + ...
also,
A(x) = 1/(1 - x*A(x)) + x*(2 - A(x))/(1 - 2*x*A(x))^2 + x^2*(2^2 - A(x))^2/(1 - 2^2*x*A(x))^3 + x^3*(2^3 - A(x))^3/(1 - 2^3*x*A(x))^3 + x^4*(2^4 - A(x))^4/(1 - 2^4*x*A(x))^4 + x^5*(2^5 - A(x))^5/(1 - 2^5*x*A(x))^6 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=Vec(sum(n=0, #A, x^n*(2^n + Ser(A))^n/(1 + 2^n*x*Ser(A))^(n+1)))); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Sequence in context: A306815 A358471 A355872 * A324306 A160710 A365269

Adjacent sequences: A307120 A307121 A307122 * A307124 A307125 A307126

Keyword

nonn

Author

Paul D. Hanna, Mar 26 2019

Status

approved