OFFSET
0,4
FORMULA
G.f. A(x) satisfies:
(1) A(x) = (1 - x*A(x)) * Sum_{n>=0} x^n / (1 - x*A(x)^(2*n+1)).
(2) A(x) = (1 - x*A(x)) * Sum_{n>=0} x^n*A(x)^n / (1 - x*A(x)^(2*n)).
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 12*x^4 + 51*x^5 + 229*x^6 + 1079*x^7 + 5288*x^8 + 26768*x^9 + 139255*x^10 + 741804*x^11 + 4035428*x^12 + ...
where
A(x)/(1 - x*A(x)) = 1/(1 - x*A(x)) + x/(1 - x*A(x)^3) + x^2/(1 - x*A(x)^5) + x^3/(1 - x*A(x)^7) + x^4/(1 - x*A(x)^9) + ...
also
A(x)/(1 - x*A(x)) = 1/(1-x) + x*A(x)/(1 - x*A(x)^2) + x^2*A(x)^2/(1 - x*A(x)^4) + x^3*A(x)^3/(1 - x*A(x)^6) + x^4*A(x)^4/(1 - x*A(x)^8) + ...
PROG
(PARI) {a(n) = my(A=1); for(i=1, n, A = (1-x*A) * sum(m=0, n, x^m / (1 - x*A^(2*m+1) +x*O(x^n)) ) ); polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = (1-x*A) * sum(m=0, n, x^m*A^m / (1 - x*A^(2*m) +x*O(x^n)) ) ); polcoeff(H=A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 25 2021
STATUS
approved