OFFSET
0,2
COMMENTS
Given g.f. A(x), x*A(x) equals a series reversion of x*G(-x) where G(x) is the g.f. of A355868.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} (2*x + (-x)^n * A(x)^n)^n.
(2) 1 = Sum_{n=-oo..+oo} -x^(2*n+1) * A(x)^(n+1) * (2 + (-x)^n * A(x)^(n+1))^n.
(3) 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^(n^2) / (1 - 2*(-x)^(n+1) * A(x)^n)^n.
(4) 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^(n^2) / (1 + 2*(-x)^(n+1) * A(x)^n)^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 5*x^2 + 13*x^3 + 30*x^4 + 74*x^5 + 202*x^6 + 616*x^7 + 2126*x^8 + 7828*x^9 + 29366*x^10 + 110398*x^11 + 414214*x^12 + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.2170550872218893465015254812376904599677836767029937...
A(1/5) = 1.8185729641608353079390837085677719656772552871159724...
PROG
(PARI) {a(n) = my(A=[1]);
for(i=1, n, A = concat(A, 0); A[#A] = polcoeff(-1 + sum(m=-#A, #A, (2*x + (-x*Ser(A))^m)^m ), #A)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 10 2023
STATUS
approved